Tuned line plot transformation

Abstract

A method of decoding can be performed by at least one processor and can include receiving an entropy encoded bitstream including compressed video data, generating one or more dequantized blocks, determining whether at least one of a height and a width of the one or more dequantized blocks is greater than or equal to a predetermined threshold, and in response to at least one of the height or the width of the one or more dequantized blocks being greater than or equal to the predetermined threshold, using a tuned look-up-table transform (LGT) kernel to transform encode the dequantized blocks to perform a direct matrix multiplication for each of a horizontal dimension and a vertical dimension of the one or more dequantized blocks.

Description

[0001] Priority information

[0002] This application claims priority to U.S. Provisional Patent Application No. 63 / 023,697, filed May 12, 2020, and U.S. Patent Application No. 17 / 105,809, filed November 27, 2020, both of which are incorporated herein by reference.

[0003] background

[0004] field

[0005] This disclosure relates to video compression techniques and inter-frame and intra-frame prediction in advanced video codecs. Specifically, this disclosure relates to a set of advanced video coding techniques designed for efficient compression of video data, such as video coding / decoding techniques other than High Efficiency Video Coding (HEVC), such as Universal Video Coding (VVC), AOMediaVideo1 (AV1), and subsequent techniques.

[0006] Introduction to existing technologies

[0007] Video encoding and decoding using inter-frame or intra-frame prediction with motion compensation has been known for decades. Uncompressed digital video can comprise a series of pictures, each with a spatial dimension of, for example, 1920×1080 luma samples and associated chroma samples. This series of pictures can have a fixed or variable picture rate (also informally referred to as the frame rate), such as 60 pictures per second or 60 Hz. Uncompressed video has very high bitrate requirements. For example, at 8 bits per sample, 1080p60 4:2:0 video (with 1920×1080 luma sample resolution at a 60 Hz frame rate) requires close to 1.5 Gbit / s of bandwidth. One hour of such video could require more than 600 GB of storage space.

[0008] One objective of video encoding and decoding is to reduce redundancy in the input video signal through compression. Compression can help reduce the aforementioned bandwidth or storage requirements, in some cases by two orders of magnitude or more. Lossless compression, lossy compression, and combinations thereof can be employed. Lossless compression refers to a technique that reconstructs an exact copy of the original signal from the compressed original signal. When using lossy compression, the reconstructed signal may differ from the original signal, but the distortion between the original and reconstructed signals is small enough that the reconstructed signal can be used for the intended application. In the case of video, lossy compression is widely used. The tolerable amount of distortion depends on the application; for example, users of some consumer streaming applications may tolerate higher distortion than users of television contribution applications. The achievable compression ratio reflects this: higher permissible / acceptable distortion results in a higher compression ratio.

[0009] Video compression / decompression technology is generally understood by those in the general technical field. Typically, to compress video or image data, a series of functional steps are performed to produce a compressed video or image file. While images such as 360° images (e.g., captured by a 360° camera) are suitable for compression, the compression of video files will be interpreted for ease of reading. To generate a compressed video file, under conventional standards (e.g., H.264, H.265), an uncompressed video sample stream received from a video source can be segmented or parsed, resulting in sample blocks of two or more reference images.

[0010] Bidirectional prediction can involve techniques in which prediction units (PUs), for example, predict sample blocks, can be predicted from two motion-compensated sample blocks of two or more reference images. Bidirectional prediction was first introduced in the video coding standard MPEG-1 and is also found in other video coding technologies and standards such as MPEG-2 Part 2 (or H.262), H.264, and H.265.

[0011] During sample reconstruction in bidirectional prediction PU, when the compressed video file is decompressed, motion-compensated and interpolated input samples from each reference block are multiplied by a weighting factor, which may be different for each reference block. These weighted sample values ​​from two reference blocks are then added to generate the samples being reconstructed. These samples can be further processed using mechanisms such as loop filtering.

[0012] In MPEG-1 and MPEG-2, weighting factors can be determined based on the relative temporal distance between the picture to which the PU being reconstructed belongs and two reference pictures. This is feasible because, in MPEG-1 and MPEG-2, one of the two reference I or P pictures is in the "past," while the other is in the "future" (in terms of presentation order) of the B picture being reconstructed, and because, in MPEG-1 and MPEG-2, a clearly defined temporal relationship is established relative to its reference pictures for any picture being reconstructed.

[0013] Starting with H.264, the concept of reference image selection for bidirectional prediction images was relaxed, so that the reference image only needs to appear earlier in the decoding order, rather than earlier in the rendering order. Furthermore, the concept of time was relaxed because neither H.264 nor H.265 requires constrained / fixed image intervals in the temporal domain. Therefore, the decoder can no longer calculate weight factors based on the temporal information available in the bitstream. Instead, H.264 and H.265 include a "default value" of 0.5 as the weight factor for the reference sample of the bidirectional prediction image. This default value can be overridden by the syntax available in the slice header, which is `pred_weight_table()`. The default value of 0.5 or the information in `pred_weight_table` (prediction_weight_table) can be applied to all bidirectional prediction PUs in a given slice.

[0014] AOMediaVideo1 (AV1), as the successor to VP9, ​​was developed by the Alliance for Open Media (AOMedia), an alliance founded in 2015 that includes semiconductor companies, video-on-demand providers, video content producers, software development companies, and web browser vendors. Many components of the AV1 project stem from previous research efforts by alliance members. Individual contributors began experimenting with technology platforms years ago: Xiph's / Mozilla's Daala released code in 2010, Google's experimental VP9 evolution project VP10 was released on September 12, 2014, and Cisco's Thor released it on August 11, 2015. Building upon the VP9 codebase, AV1 incorporates additional technologies, some of which were developed using these experimental formats. The first version 0.1.0 of the AV1 reference codec was released on April 7, 2016. The alliance announced the release of the AV1 bitstream specification, along with software-based reference encoders and decoders, on March 28, 2018. On June 25, 2018, a validation version 1.0.0 of the specification was released. On January 8, 2019, a validation version 1.0.0 of the specification with errata table 1 was released. The AV1 stream specification includes a reference video codec.

[0015] Currently, AOMediaVideo2 (AV2) is under development, and an 8-bit / 10-bit conversion core has been designed for it.

[0016] List of existing technologies

[0017] Non-patent literature 1: HE Egilmez, E. Pavez, A. Ortega, “Graph learning from data under Laplacian and structural constraints”, IEEE Journal of Selected Topics in Signal Processing., vol. 11, no. 6, Sept. 2017.

[0018] Non-patent literature 2: HE Egilmez, YH Chao, A. Ortega, B. Lee, and S. Yea, “GBST: Separable transforms based on line graphs for predictive video coding,” 2016 IEEE International Conference on Image Processing (ICIP), Sept 2016, pp.2375–2379.

[0019] Non-Patent Literature 3: HE Egilmez, YH Chao, and A. Ortega, “Graph-based transforms for video coding,” CoRR, vol. abs / arXiv:1909.00952, 2019.

[0020] Non-Patent Document 4: M. Zhou, Y. Hu, “CE6-related: A study of primary transforms,” Joint Video Experts Team (JVET) of ITU-T SG 16 WP 3 and ISO / IEC JTC 1 / SC 29 / WG 11, JVET-M0046, 13th Meeting: Marrakesh, MA, 9–18 Jan. 2019.

[0021] The problem to be solved

[0022] This disclosure relates to a set of advanced video coding techniques designed for efficient compression of video data. Summary of the Invention

[0023] More specifically, one aspect of this disclosure relates to methods, apparatus, and computer-readable media that perform one or more of the following: (a) methods for tuning line graph transforms (LGTs) of different sizes; (b) 8-bit LGT transform kernels of different sizes (4-point, 8-point, 16-point, 32-point, and 64-point) tuned using the method; and (c) methods for enabling 32-point and 64-point LGTs.

[0024] According to one aspect of this disclosure, a decoding method may be executed by at least one processor and may include: receiving an entropy-coded bitstream comprising compressed video data; generating one or more dequantized blocks through a process comprising: parsing the bitstream; decoding the entropy-coded bitstream; and dequantizing each of the one or more blocks; determining whether at least one of the height and width of the one or more dequantized blocks is greater than or equal to a predetermined threshold; and in response to the at least one of the height or width of the one or more dequantized blocks being greater than or equal to the predetermined threshold, performing transform encoding on the dequantized blocks using a tuned line graph transform (LGT) kernel to perform direct matrix multiplication on each of the horizontal and vertical dimensions of the one or more dequantized blocks.

[0025] The predetermined threshold can have a value corresponding to either 32 or 64.

[0026] The predetermined threshold can have a value corresponding to one of 4, 8, 16, 32, 64, 128, and 256.

[0027] The decoding method may further include: generating an LGT kernel using at least one of a first self-loop weight or a second self-loop weight before using the LGT kernel, wherein the first self-loop weight and the second self-loop weight are each pre-set to specific corresponding values.

[0028] According to one embodiment, the tuned LGT kernel is a discrete cosine transform (DCT) or a discrete sine transform (DST), which is derived from DCT-2, DCT-8, or DST-7 derived from a generalized graph Laplacian (GGL) matrix of a certain form.

[0029] The decoding method may further include at least one of the following, and may include all or some of the following: deriving DCT-2 by setting the first self-loop weight to 0; deriving DST-7 by setting the first self-loop weight to a first predetermined value; deriving DCT-8 by setting the second self-loop weight to the first predetermined value; deriving DST-4 by setting the first self-loop weight to twice the first predetermined value; and deriving DCT-4 by setting the second self-loop weight to twice the first predetermined value.

[0030] Tuning an LGT kernel may include: tuning LGT kernels of different sizes using first self-loop weights and second self-loop weights to derive a first transform kernel; and brute-force tuning one or more individual weights of one or more basis vectors of the first transform kernel to derive a second transform kernel, wherein the second transform kernel is a tuned LGT kernel.

[0031] The decoding method may further include: determining whether Np LGT is enabled; and when Np LGT is enabled, using only Discrete Cosine Transform (DCT)-2 and LGT as candidates for the transform type of the edge of length N of the dequantized block.

[0032] The decoding method may further include: determining whether Np LGT is enabled; and when Np LGT is enabled, using only DCT-2, LGT, and flipped LGT as candidates for the transformation type of the edge of length N of the dequantized block.

[0033] The decoding method may further include: determining whether Np LGT is enabled; and when Np LGT is enabled, avoiding the use of identity transformations on the edges of the dequantized block of length N.

[0034] The decoding method may further include: determining whether Np LGT is enabled; and when Np LGT is enabled, using supported transform types for mutually different intra-frame prediction residuals and inter-frame prediction residuals.

[0035] The decoding method may further include: for intra-frame prediction residuals, supported transform types include DCT-2 and LGT.

[0036] The decoding method may further include: for inter-frame prediction residuals, supported transform types include DCT-2, LGT, and inverted LGT.

[0037] The decoding method may further include: generating a displayable image based at least in part on the transformed dequantized block; and performing control to display the displayable image via a display.

[0038] According to one embodiment, a non-transitory computer-readable storage medium stores instructions that, when executed, cause a system or device including one or more processors to: receive an entropy-coded bitstream comprising compressed video data; generate one or more dequantized blocks through a process comprising: parsing the bitstream; decoding the entropy-coded bitstream; and dequantizing each of the one or more blocks; determining whether at least one of the height and width of the one or more dequantized blocks is greater than or equal to a predetermined threshold; and, in response to the height or width of the one or more dequantized blocks being greater than or equal to the predetermined threshold, perform transform encoding on the dequantized blocks using a tuned line graph transform (LGT) core to perform direct matrix multiplication on each of the horizontal and vertical dimensions of the one or more dequantized blocks.

[0039] The instructions may be further configured to cause a system or device including one or more processors to: generate a displayable image based at least in part on the transformed dequantized block; and perform control to display the displayable image via a display.

[0040] According to one embodiment, an apparatus may include: at least one memory configured to store computer program code; and at least one processor configured to access the at least one memory and operate according to the computer program code, the computer program code including: receiving code configured to cause the at least one processor to receive an entropy-coded bitstream comprising compressed video data; generating code configured to cause the at least one processor to generate one or more dequantized blocks through a process including: parsing the bitstream, decoding the entropy-coded bitstream, and dequantizing each of the one or more blocks; determining code configured to cause the at least one processor to determine whether at least one of the height and width of the one or more dequantized blocks is greater than or equal to a predetermined threshold; and transform encoding code configured to cause the at least one processor to: in response to the height and width of the one or more dequantized blocks being greater than or equal to the predetermined threshold, transform-encode the dequantized blocks using a tuned line graph transform (LGT) core to perform direct matrix multiplication on each of the horizontal and vertical dimensions of the one or more dequantized blocks.

[0041] The apparatus may further include a display, wherein the computer program code may further include display code, the display code being configured to cause at least one processor to: generate a displayable image based at least in part on the transformed dequantized block; and perform control to display the displayable image via the display. Attached Figure Description

[0042] Further features, properties, and various advantages of the disclosed subject matter will become more apparent from the following detailed description and accompanying drawings, in which:

[0043] Figure 1 This is a simplified block diagram of a communication system according to one embodiment.

[0044] Figure 2 This is a simplified block diagram of a streaming system according to one embodiment.

[0045] Figure 3 This is a simplified block diagram of a decoder according to one embodiment.

[0046] Figure 4 This is a simplified block diagram of an encoding system including an encoder and a local decoder according to one embodiment.

[0047] Figure 5A Table 1, according to one embodiment, shows the AV1 master transform basis functions (DCT-2, DST-4, and DST-7 for N-point inputs).

[0048] Table 2 ( Figure 5B The diagram illustrates a planar rotation (butterfly and matrix representation) according to one embodiment.

[0049] Figure 6A and Figure 6B A partial butterfly lookup table used in AV1 according to one embodiment is shown, which can be used to generate values ​​of trigonometric functions.

[0050] Figure 7A A general LGT characterized by self-loop weights and edge weights is shown according to an example embodiment.

[0051] Figure 7B A matrix Lc is shown according to an example embodiment.

[0052] Figure 8 This is a schematic diagram of a computer system according to one embodiment.

[0053] Figure 9 This is a schematic diagram of an exemplary method according to one embodiment. Detailed Implementation

[0054] In the implementation of this application, the collection and processing of relevant data should be strictly in accordance with the requirements of relevant laws and regulations, obtain the informed consent or separate consent of the personal information subject, and carry out subsequent data use and processing within the scope of laws and regulations and the authorization of the personal information subject.

[0055] Figure 1 A simplified block diagram of a communication system (100) according to an embodiment of the present disclosure is shown. The system (100) may include at least two terminals (110, 120) interconnected via a network (150). For one-way data transmission, the first terminal (110) may encode video data at a local location for transmission over the network (150) to the other terminal (120). The second terminal (120) may receive the encoded video data from the other terminal from the network (150), decode the encoded data, and display the recovered video data. One-way data transmission can be common in applications such as media services.

[0056] Figure 1 A second pair of terminals (130, 140) configured to support bidirectional transmission of encoded video is shown, which may occur, for example, during a video conference. For bidirectional data transmission, each terminal (130, 140) may encode video data acquired at a local location for transmission over a network (150) to the other terminal. Each terminal (130, 140) may also receive encoded video data transmitted by the other terminal, decode the encoded data, and display the recovered video data on a local display device.

[0057] exist Figure 1In this disclosure, terminals (110-140) may be shown as laptop computer 110, server 120, and smartphones 130 and 140, but the principles of this disclosure are not limited thereto. Embodiments of this disclosure are found to be applied on other devices, including but not limited to laptop computers, tablets, media players, and / or dedicated video conferencing equipment. Network (150) refers to any number of networks that transmit encoded video data between terminals (110-140), including, for example, wired and / or wireless communication networks. Communication networks (150) may exchange data in circuit-switched and / or packet-switched channels. Representative networks include telecommunications networks, local area networks (LANs), wide area networks (WANs), and / or the Internet. For the purposes of this discussion, unless explained below, the architecture and topology of the network (150) may be irrelevant to the operation of this disclosure.

[0058] As an example of the application of the disclosed topic, Figure 2 The placement of decoder 210 in the streaming environment / streaming system 200 is shown. (Reference) Figure 3 and Figure 4 Decoder 433 further discusses decoder 210. Decoder 210 may correspond to... Figure 3 decoder 210 or Figure 4 Decoder 433 in the middle.

[0059] The disclosed topics are equally applicable to other video-enabled applications, including video conferencing, digital TV, storing compressed video on digital media such as CDs, DVDs, and Memory Sticks.

[0060] like Figure 2As shown, the streaming system (200) may include an acquisition subsystem (213) that may include a video source (201) such as a digital camera, which may create an uncompressed video sample stream (202). The video sample stream (202), depicted as thick lines to emphasize its high data volume compared to the encoded video stream, may be processed by a source encoder (203) coupled to the camera (201). The source encoder (203) may include hardware (e.g., a processor or circuitry and memory), software, or a combination of hardware and software to implement or enforce aspects of the disclosed subject matter as described in more detail below. The encoded video stream (204), depicted as thin lines to emphasize its lower data volume compared to the sample stream, may be stored on a streaming server (205) for future use. One or more streaming clients (206, 208) may access the streaming server (205) to retrieve a copy (207, 209) of the encoded video stream (204). The client (206) may include a video decoder (210) that can decode an incoming copy (207) of an encoded video stream and produce an output video sample stream (211) that can be displayed on a display (212) or another presentation device.

[0061] Figure 3 A functional block diagram of a decoder (210) (e.g., a video decoder) according to one embodiment of the present disclosure is shown. Figure 3 As shown, the receiver (310) can receive one or more encoded video sequences to be decoded by the decoder (210). In the same embodiment or another embodiment, one encoded video sequence can be received at a time, wherein the decoding of each encoded video sequence is independent of the decoding of other encoded video sequences. The encoded video sequences can be received from a channel (312), which can be a hardware / software link to a storage device storing the encoded video data. The receiver (310) can receive encoded video data, as well as other data, such as encoded audio data and / or auxiliary data streams, that can be forwarded to their respective user entities (not depicted). The receiver (310) can separate the encoded video sequences from other data. To prevent network jitter, a buffer memory (315) can be coupled between the receiver (310) and the entropy decoder / parser (320) (hereinafter referred to as the "parser"). When the receiver (310) receives data from a storage / forwarding device with sufficient bandwidth and controllability or from an isochronous synchronization network, the buffer (315) may not be required, or the buffer may be made smaller. In order to make the best use of business packet networks such as the Internet, a buffer (315) may be required, which can be relatively large and can advantageously have an adaptive size.

[0062] The video decoder (210) may include a parser (310) to reconstruct symbols (421) from the entropy-encoded video sequence. These symbols include information for managing the operation of the decoder (210), and potential information for controlling a display device (212) (e.g., a display), which is not part of the decoder but may be coupled to it, such as... Figure 2 and 3 As shown in the figure. The control information for the display device may be a parameter set fragment (not shown) of Supplemental Enhancement Information (SEI message) or Video Usability Information (VUI). The parser (320) may parse / entropy decode the received encoded video sequence. The encoding of the encoded video sequence may be based on video coding techniques or standards and may follow principles well known to those skilled in the art, including variable-length coding, Huffman coding, arithmetic coding with or without context sensitivity, etc. The parser (320) may extract a subgroup parameter set of at least one subgroup of pixels in the subgroup of pixels for use in the video decoder based on at least one parameter corresponding to a group. The subgroup may include a Group of Pictures (GOP), picture, tile, slice, macroblock, coding unit (CU), block, transform unit (TU), prediction unit (PU), etc. The entropy decoder / parser 320 can also extract information from encoded video sequences, such as transform coefficients, quantizer parameter values, motion vectors, and so on.

[0063] The parser (320) performs entropy decoding / parsing operations on the video sequence received from the buffer (315) to create symbols (321). The reconstruction of symbols (321) may involve multiple different units, depending on the type of encoded video pictures or portions thereof (e.g., inter-frame and intra-frame pictures, inter-frame and intra-frame blocks) and other factors. Which units are involved and how they are involved can be controlled by subgroup control information parsed by the parser (320) from the encoded video sequence. For brevity, the flow of such subgroup control information between the parser (320) and the various units described below is not described.

[0064] In addition to the functional blocks already mentioned, decoder 210 can be conceptually subdivided into several functional units as described below. In practical embodiments operating under commercial constraints, many of these units interact closely with each other and can be integrated with one another. However, for the purposes of describing the disclosed subject matter, it is appropriate to conceptually subdivide them into the functional units described below.

[0065] The first unit can be a scaler / inverse transform unit (351). The scaler / inverse transform unit (351) can receive quantization transform coefficients as symbols (321) and control information from the parser (320), including which transform mode to use, block size, quantization factor, quantization scaling matrix, etc. The scaler / inverse transform unit can output a block containing sample values, which can be input into the aggregator (355).

[0066] In some cases, the output samples of the scaler / inverse transform unit (351) may belong to intra-coded blocks; that is, blocks that do not use predictive information from previously reconstructed images, but can use predictive information from previously reconstructed portions of the current image. Such predictive information may be provided by the intra-picture prediction unit (352). In some cases, the intra-picture prediction unit (352) uses surrounding reconstructed information extracted from the current (partially reconstructed) image (358) to generate blocks of the same size and shape as the blocks being reconstructed. In some cases, the aggregator (355) adds the predictive information generated by the intra-picture prediction unit (352) to the output sample information provided by the scaler / inverse transform unit (351) based on each sample.

[0067] In other cases, the output samples of the scaler / inverse transform unit (351) may belong to inter-frame coded blocks and latent motion compensation blocks. In this case, the motion compensation prediction unit (353) may access the reference image memory (357) to extract samples for prediction. After motion compensation is performed on the extracted samples according to the symbol (321), these samples may be added by the aggregator (355) to the output of the scaler / inverse transform unit (referred to in this case as residual samples or residual signals) to generate output sample information. The motion compensation unit's retrieval of predicted samples from the address in the reference image memory may be controlled by motion vectors, and these motion vectors are available to the motion compensation unit in the form of the symbol (321), which, for example, includes X, Y, and reference image components. Motion compensation may also include interpolation of sample values ​​extracted from the reference image memory when using subsample precise motion vectors, motion vector prediction mechanisms, etc.

[0068] The output samples of the aggregator (355) can be employed by various loop filtering techniques in the loop filter unit (356). Video compression techniques may include in-loop filtering techniques controlled by parameters included in the encoded video bitstream, and these parameters can be used as symbols (321) from the parser (320) in the loop filter unit (356). However, in other embodiments, the video compression techniques may also respond to metadata obtained during decoding of a previous (in decoding order) portion of the encoded picture or encoded video sequence, and to previously reconstructed and loop-filtered sample values.

[0069] The output of the loop filter unit (356) can be a sample stream, which can be output to the display device (212) and stored in the reference image memory (357) for subsequent inter-frame image prediction.

[0070] Once fully reconstructed, some of the encoded images can be used as reference images for future predictions. Once the encoded images have been fully reconstructed and the encoded images (by, for example, the parser (320)) are identified as reference images, the current reference image (358) can become part of the reference image buffer (357), and new current image memory can be reallocated before the reconstruction of subsequent encoded images begins.

[0071] The video decoder 210 can perform decoding operations according to a predetermined video compression technique that can be archived, for example, in the ITU-T H.265 standard. The encoded video sequence may conform to the syntax specified by the video compression technique or standard used, in the sense that the encoded video sequence follows the syntax of the video compression technique or standard and the configuration file recorded in the video compression technique or standard. For compliance, the complexity of the encoded video sequence is also required to be within the range defined by the hierarchy of the video compression technique or standard. In some cases, the hierarchy limits the maximum picture size, maximum frame rate, maximum reconstruction sampling rate (measured in, for example, megasamples per second), and / or maximum reference picture size. In some cases, the limitations set by the hierarchy can be further limited by the Hypothetical Reference Decoder (HRD) specification and the metadata managed by the HRD buffer, which is represented by signals in the encoded video sequence.

[0072] In this embodiment, the receiver (310) may receive supplemental (redundant) data along with the encoded video. This supplemental data may be a portion of the encoded video sequence. The supplemental data may be used by the video decoder (210) to properly decode the data and / or more accurately reconstruct the original video data. The supplemental data may take the form of, for example, temporal, spatial, or signal-to-noise ratio (SNR) enhancement layers, redundant slices, redundant images, forward error correction codes, etc.

[0073] Figure 4 This may be a functional block diagram of an encoding system (400) including a source encoder (203) according to an embodiment of the present disclosure, wherein the source encoder (203) may be a video encoder (203).

[0074] The encoder (203) can receive video samples from a video source (201) (which is not part of the encoder), which can capture video images that will be encoded by the encoder (203).

[0075] A video source (201) can provide a sequence of source video samples in the form of a digital video sample stream to be encoded by an encoder (203). This digital video sample stream can have any suitable bit depth (e.g., 8-bit, 10-bit, 12-bit…), any color space (e.g., BT.601 YCrCb, RGB…), and any suitable sampling structure (e.g., YCrCb 4:2:0, YCrCb 4:4:4). In a media service system, the video source (201) can be a storage device storing previously prepared video, which may include memory and a processor. In a video conferencing system, the video source (201) can include a camera that captures local image information as a video sequence. Video data can be provided as multiple individual pictures, which are given motion when viewed sequentially. The pictures themselves can be constructed as spatial pixel arrays, where each pixel may include one or more samples, depending on the sampling structure, color space, etc., used. Those skilled in the art can easily understand the relationship between pixels and samples. The following focuses on describing samples.

[0076] According to an embodiment, the encoder (203) can encode and compress images of a source video sequence into an encoded video sequence in real time or under any other time constraints required by the application. Implementing an appropriate encoding rate is a function of the controller (450). The controller 450 can control and is functionally coupled to other functional units described below. For simplicity, coupling is not shown in the figures. Parameters set by the controller 450 may include rate control related parameters (image skipping, quantizer, λ value of rate-distortion optimization techniques, etc.), image size, group of pictures (GOP) layout, maximum motion vector search range, etc. Those skilled in the art will readily recognize other functions of the controller (150) related to a video encoder (203) optimized for a particular system design.

[0077] Some video encoders operate within an “encoding loop” readily recognized by those skilled in the art. In a simplified description, the encoding loop may include: an encoding portion of a source encoder (430) (e.g.), responsible for creating symbols based on the input image to be encoded and a reference image; and a local decoder (433) embedded in encoder 203, which reconstructs the symbols to create sample data in a manner similar to how the remote decoder 210 creates sample data (since any compression between the symbols and the encoded video stream is lossless in the video compression techniques considered in this application). The reconstructed sample stream is input to a reference image memory (434). Since decoding of the symbol stream produces bit-precise results independent of the decoder location (local or remote), the contents of the reference image buffer are also bit-precisely corresponding between the local and remote encoders. In other words, the reference image samples “seen” by the encoder’s prediction portion are exactly the same sample values ​​that the decoder will “see” during prediction. This fundamental principle of reference image synchronization (and the drift that occurs, for example, due to channel errors) is well known to those skilled in the art.

[0078] The operation of the “local” decoder (433) can be combined with, for example, the above-mentioned Figure 3 The "remote" decoder (210) described in detail is the same. However, another brief reference is provided. Figure 3 When symbols are available and the entropy encoder (445) and parser (320) are able to encode / decode the symbols into an encoded video sequence without loss, the entropy decoding portion of the decoder 210 (including the channel (312), receiver (310), buffer (315) and parser (320)) may not be fully implemented in the local decoder (433).

[0079] It can then be observed that any decoder technique other than parsing / entropy decoding, which exists in the decoder, must also exist in the corresponding encoder in essentially the same functional form. The encoder technique is the inverse of the described decoder technique. More detailed descriptions are needed only in certain areas, and are provided below.

[0080] As part of its operation, the source encoder (430) performs motion-compensated predictive coding. This motion-compensated predictive coding predictively encodes the input frame, referencing one or more previously encoded frames in the video sequence designated as "reference frames." In this manner, the encoding engine (432) encodes the differences between pixel blocks of the input frame and pixel blocks of the reference frame, which can be selected as a prediction reference for the input frame.

[0081] The local video decoder (433) can decode encoded video data of a frame that can be designated as a reference frame, based on symbols created by the source encoder (430). The operation of the encoding engine (432) can be a lossy process. When the encoded video data can be decoded by the video decoder (430), Figure 4 When the source video sequence (not shown) is decoded, the reconstructed video sequence can typically be a copy of the source video sequence with some errors. The local video decoder (433) replicates the decoding process, which can be performed by the video decoder on the reference frame, and allows the reconstructed reference image to be stored in the reference frame cache (434). In this way, the source encoder (203) can locally store a copy of the reconstructed reference frame that shares the same content (no transmission errors) as the reconstructed reference frame that will be obtained by the remote video decoder.

[0082] The predictor (435) can perform a prediction search against the encoding engine (432). That is, for a new frame to be encoded, the predictor (435) can search in the reference image memory (434) for sample data (as candidate reference pixel blocks) or certain metadata, such as reference image motion vectors, block shapes, etc., that can serve as appropriate prediction references for the new image. The predictor (435) can operate pixel-by-pixel based on the sample blocks to find suitable prediction references. In some cases, based on the search results obtained by the predictor (435), it can be determined that the input image can have prediction references obtained from multiple reference images stored in the reference image memory (434).

[0083] A controller (450) that may include a processor may manage the encoding operations of the source encoder (203), including, for example, setting parameters and subgroup parameters for encoding video data.

[0084] The outputs of all the above-mentioned functional units can be entropy encoded in the entropy encoder (445). The entropy encoder performs lossless compression on the symbols generated by the various functional units according to techniques known to those skilled in the art, such as Huffman coding, variable-length coding, and arithmetic coding, thereby converting the symbols into an encoded video sequence.

[0085] The transmitter (440) can buffer the encoded video sequence created by the entropy encoder (445) in preparation for transmission via a communication channel (460), which may be a hardware / software link to a storage device that will store the encoded video data. The transmitter (440) can combine the encoded video data from the video encoder (203) with other data to be transmitted, such as encoded audio data and / or auxiliary data streams (source not shown).

[0086] The controller (450) manages the operation of the encoder (203). During encoding, the controller (450) can assign a specific encoded image type to each encoded image, but this may affect the encoding techniques applicable to the corresponding images. For example, images can typically be assigned to any of the following frame types, selected from, for example, intra-frame images (I-images), predictive images (P-images), and bidirectional predictive images (B-images).

[0087] An intra-frame picture (I-picture) is a picture that can be encoded and decoded without using any other frames in the sequence as a prediction source. Some video codecs allow different types of intra-frame pictures, including, for example, Independent Decoder Refresh pictures. Those skilled in the art are familiar with variations of I-pictures and their corresponding applications and characteristics.

[0088] A predictive picture (P-picture) can be a picture that can be encoded and decoded using intra-frame prediction or inter-frame prediction, which uses at most one motion vector and reference index to predict sample values ​​for each block.

[0089] Bidirectional predictive images (B-images) can be images that can be encoded and decoded using intra-frame or inter-frame prediction, which uses at most two motion vectors and a reference index to predict sample values ​​for each block. Similarly, multiple predictive images can use more than two reference images and associated metadata to reconstruct a single block.

[0090] Source images are typically spatially subdivided into multiple sample blocks (e.g., 4×4, 8×8, 4×8, or 16×16 sample blocks), and encoded block by block. These blocks can be predictively coded with reference to other (already coded) blocks, determined by the coding assignment of the corresponding images applied to the block. For example, a block of an I-image can be non-predictively coded, or it can be predictively coded with reference to already coded blocks of the same image (spatial prediction or intra-frame prediction). A pixel block of a P-image can be predictively coded with reference to a previously coded reference image via spatial prediction or temporal prediction. A block of a B-image can be predictively coded with reference to one or two previously coded reference images via spatial prediction or temporal prediction.

[0091] The source encoder (203) can perform encoding operations according to, for example, a predetermined video coding technique or standard of ITU-T H.265 or VCC. In operation, the source encoder (203) can perform various compression operations, including predictive coding operations that utilize temporal and spatial redundancy in the input video sequence. Therefore, the encoded video data can conform to the syntax specified by the video coding technique or standard used.

[0092] In this embodiment, the transmitter (440) may transmit additional data while transmitting encoded video. The source encoder (530) may include such data as part of the encoded video sequence. The additional data may include other forms of redundant data such as temporal / spatial / SNR enhancement layers, redundant pictures and slices, auxiliary enhancement information (SEI) messages, video availability information (VUI) parameter set fragments, etc.

[0093] As discussed above, the encoding of encoded video sequences can be performed according to video coding techniques or standards, and can follow various principles well known to those skilled in the art, including variable-length coding, Huffman coding, and arithmetic coding with or without context sensitivity. In some streaming systems, the video bitstream (204, 207, 209) can be encoded according to certain video coding / compression standards. Examples of these standards include ITU-T HEVC Recommendation H.265.

[0094] AV1 main converter

[0095] To support extended coded block segmentation in AOMediaVideo1 (AV1), a variety of transform sizes (ranging from 4 to 64 points for each dimension) and transform shapes (square; rectangles with width / height ratios of 2:1 / 1:2 and 4:1 / 1:4) are introduced in AV1.

[0096] In AV1, a single (1D) transformation includes: (a) for DCT-2, 4-point (4p), 8-point (8p), 16-point (16p), 32-point (32p), and 64-point (64p); (b) for asymmetric DST and its inverted version, 4p, 8p, and 16p; and (c) for identity transformation, 4p, 8p, 16p, and 32p. The 2D transformation process may involve the use of hybrid transformations (e.g., different transformations for each dimension of the encoded residual block), where the selection of the transformation for each dimension is based on the rate-distortion (RD) criterion. The basis functions of DCT-2, DST-4, and DST-7 used in AV1 are listed in Table 1 (…). Figure 5A Listed in ).

[0097] The basis functions of DCT-2 and asymmetric DST used in AV1 are as follows: Figure 5A The following are listed in Table 1.

[0098] Figure 5A Table 1 shows the AV1 master transform basis functions (DCT-2, DST-4, and DST-7 for N-point inputs).

[0099]

[0100]

[0101] The DCT-2 (4p to 64p), DST-4 (8p, 16p), and DST-7 (4p) transforms exhibit symmetric / antisymmetric properties. Therefore, a so-called "partial butterfly" implementation is supported to reduce the number of operation counts (e.g., one or more of multiplication, addition / subtraction, and shifting). According to one embodiment, a usable partial butterfly implementation may involve planar rotations at various angles using trigonometric cosine and trigonometric sine functions, such as... Figure 5B As described.

[0102] Figure 5B Planar rotation (butterfly and matrix representation) according to one embodiment is shown.

[0103] According to one embodiment, such as Figure 6A and Figure 6B The 12-bit lookup table shown can be used to generate values ​​for trigonometric functions. Figure 6A A partial butterfly lookup table of DCT-2 (4p to 64p) / DST-4 (8p, 16p) used in AV1 according to one embodiment is shown. Figure 6B A partial butterfly lookup table of DST-7 (4p) used in AV1 according to one embodiment is shown.

[0104] AV2 main converter

[0105] During the ongoing development of AV2, Line Graph Transformation (LGT) is introduced to replace the aforementioned 1D DST.

[0106] A graph is a general mathematical structure consisting of a set of vertices and edges, which can be used to model the relationships between objects of interest. See, for example, Non-Patent Literature 1. In practice, weighted graphs (where a set of weights is assigned to edges and possibly to vertices) can provide a sparse representation for robust modeling of signals / data. LGTs can improve coding efficiency by providing a better fit to different block statistics. In some examples (e.g., Non-Patent Literatures 2 and 3), separable LGTs are designed and optimized by learning line graphs from data (see...). Figure 7A The underlying row-by-row and column-by-column statistics of the block residual signal are modeled using the associated generalized graph Laplacian (GGL) matrix to derive the LGT.

[0107] For example, Figure 7A A general LGT characterized by self-loop weights vc1, vc2 and edge weights wc is shown.

[0108] According to one embodiment, such as Figure 7AAs shown, given a weighted graph G(W, V), the GGL matrix can be defined as:

[0109] L C = D - W + V (Equation 1)

[0110] Where W is composed of non-negative edge weights w c The adjacency matrix is ​​formed, where D is the angle matrix and V represents the weighted self-loop v. c1 v c2 A diagonal matrix. For example... Figure 7A As shown, matrix Lc can be represented as:

[0111]

[0112] Then, the LGT can be derived through the feature decomposition of GGL Lc.

[0113] Lc = UΦU T (Equation 3)

[0114] In this matrix, the columns of the orthogonal matrix U are the basis vectors of the LGT, and Φ is the diagonal eigenvalue matrix.

[0115] In fact, DCT and DST (including DCT-2, DCT-8, and DST-7) can be LGTs derived from a certain form of GGL, for example:

[0116] By setting v c1 = 0 and thus DCT-2 is derived (Equation 4)

[0117] By setting v c1 = w c DST-7 is derived (Equation 5).

[0118] By setting v c2 = w c And the DCT-8 is derived (Equation 6).

[0119] By setting v c1 = 2 w c DST-4 is derived (Equation 7).

[0120] By setting v c2 = 2 w c And DCT-4 is derived (Equation 8).

[0121] In AV2, LGT can be implemented as matrix multiplication. This is achieved by setting [the appropriate function] in Lc. v c1 = 2 w c Exporting a 4p LGT core means the LGT core is a DST-4. This is achieved by setting [the appropriate setting] in Lc. v c1 = 1.5 w c Exporting the 8p LGT core is achieved by setting [the appropriate configuration in Lc]. v c1 = w c The export of 16p and 32p LGT cores means that the LGT core is DST-7.

[0122] Additional tuning

[0123] In AV2, the LGT kernel can be further tuned to achieve better coding efficiency or accuracy. For example, a 64-point (64p) LGT can be introduced in AV2 to provide additional coding efficiency. Since both the 32-point (32p) LGT and the 64p LGT are computationally intensive, constraints on their application can be formulated based on one or more of the block size, supported transform types, and / or prediction modes without affecting coding efficiency.

[0124] The methods described below can be used individually or in any combination in any order. As detailed above, this can be achieved by applying self-loop weights (e.g., self-loop weights). v c1 and v c2 Different LGT kernels are generated by setting specific values. Furthermore, the basis vectors of the LGT kernel can be tuned to have orthogonality (this process involves brute-force tuning of the individual weights of the basis vectors within the dynamic range). Therefore, according to one embodiment, an efficient LGT tuning method can combine two degrees of freedom (e.g., self-loop weights). v c1 and v c2 ) and the basis vector tuning process.

[0125] According to one embodiment, the LGT tuning method may include: (a) in two degrees of freedom (e.g., self-loop weights) v c1 and v c2 (a) Tune LGT kernels of different sizes to reach an initial (first) transform kernel; and (b) further tune the individual weights of the basis vectors of the initial transform kernel by brute force to reach a final (second) transform kernel.

[0126] The two-step tuning process described above can be used to generate a set of optimal transform kernels of different sizes, which can be introduced into AV2 to achieve better coding efficiency. The transform kernels generated using this method are listed in the following sections.

[0127] When using LGT, a flipped version of LGT (i.e., a flipped LGT) can also be used. It refers to using The LGT's flip (left-right flip, or up-down flip) transformation matrix is ​​used as a transformation matrix transformation. .

[0128] Enable Np LGT

[0129] In this embodiment, 32p and 64p LGT will be referred to as Np LGT, where N can take values ​​of 32 and 64 respectively.

[0130] It proposes enabling Np LGT when there are some limitations in terms of block size and supported transformation types.

[0131] In one embodiment, Np LGT is enabled only for block sizes of W×N and N×H, where W and / or H are greater than or equal to a given threshold. Example values ​​for the threshold include 128, 64, 32, 16, 8, and 4.

[0132] In one embodiment, when Np LGT is enabled, only DCT-2 and LGT can be used as transformation type candidates for edges of length N.

[0133] In one embodiment, when Np LGT is enabled, only DCT-2, LGT, and Flipped LGT are available as transformation type candidates for edges of length N.

[0134] In one embodiment, when Np LGT is enabled, the identity transformation cannot be used for edges of length N.

[0135] In one embodiment, when Np LGT is enabled, the supported transform types differ for intra-frame prediction residuals and inter-frame prediction residuals.

[0136] In one example, for intra-frame prediction residuals, supported transform types include DCT-2 and LGT. For inter-frame prediction residuals, supported transform types include DCT-2, LGT, and inverted LGT.

[0137] refer to Figure 9 In a first simplified example, according to one aspect of this disclosure, a method for decoding a bitstream consisting of video data may be performed by at least one processor and may include: receiving an entropy-encoded bitstream comprising compressed video data (operation 601). The bitstream may refer to encoded video or images received by a decoder.

[0138] The method may further include generating one or more dequantized blocks (operation 602). Operation 602 may include generating one or more dequantized blocks through a process that includes parsing the bitstream, decoding the entropy-coded bitstream, and dequantizing each block (operation 602).

[0139] The method may further include: determining whether at least one of the height and width of the dequantized block is greater than or equal to a predetermined threshold (operation 603).

[0140] The method may further include: in response to at least one of the height and width of the dequantized block being greater than or equal to a predetermined threshold: performing transform encoding on the dequantized block using a tuned line graph transform (LGT) kernel to perform direct matrix multiplication (operation 604) on each of the horizontal and vertical dimensions of the dequantized block with reduced complexity.

[0141] The method may further include: based on the fact that at least one of the height or width of the determined dequantized block is not greater than or equal to a predetermined threshold: performing transform encoding on the original block using a Line Graph Transform (LGT) kernel to perform direct matrix multiplication (operation 605) on each of the horizontal and vertical dimensions of the original block.

[0142] The predetermined threshold can have a value corresponding to either 32 or 64.

[0143] The predetermined threshold can have a value corresponding to one of 4, 8, 16, 32, 64, 128, and 256.

[0144] The decoding method may further include: generating an LGT kernel using at least one of a first self-loop weight or a second self-loop weight before using the LGT kernel, wherein the first self-loop weight and the second self-loop weight are each pre-set to specific corresponding values.

[0145] According to one embodiment, the tuned LGT core can be a Discrete Cosine Transform (DCT) or a Discrete Sine Transform (DST), which is derived from DCT-2, DCT-8, or DST-7 from a generalized graph Laplace (GGL) matrix of a certain form.

[0146] The decoding method may further include at least one of the following, and may include all or some of the following: deriving DCT-2 by setting the first self-loop weight to 0; deriving DST-7 by setting the first self-loop weight to a first predetermined value; deriving DCT-8 by setting the second self-loop weight to the first predetermined value; deriving DST-4 by setting the first self-loop weight to twice the first predetermined value; and deriving DCT-4 by setting the second self-loop weight to twice the first predetermined value.

[0147] Tuning an LGT kernel may include: tuning LGT kernels of different sizes using first self-loop weights and second self-loop weights to derive a first transform kernel; and brute-force tuning one or more individual weights of one or more basis vectors of the first transform kernel to derive a second transform kernel, wherein the second transform kernel is a tuned LGT kernel.

[0148] The decoding method may further include: determining whether Np LGT is enabled; and when Np LGT is enabled, using only Discrete Cosine Transform (DCT)-2 and LGT as candidates for the transform type of the edge of length N of the dequantized block.

[0149] The decoding method may further include: determining whether Np LGT is enabled; and when Np LGT is enabled, using only DCT-2, LGT, and flipped LGT as candidates for the transformation type of the edge of length N of the dequantized block.

[0150] The decoding method may further include: determining whether Np LGT is enabled; and when Np LGT is enabled, avoiding the use of identity transformations on the edges of the dequantized block of length N.

[0151] The decoding method may further include: determining whether Np LGT is enabled; and when Np LGT is enabled, using supported transform types for mutually different intra-frame prediction residuals and inter-frame prediction residuals.

[0152] The decoding method may further include: for intra-frame prediction residuals, supported transform types include DCT-2 and LGT.

[0153] The decoding method may further include: for inter-frame prediction residuals, supported transform types include DCT-2, LGT, and inverted LGT.

[0154] The decoding method may further include: generating a displayable image based at least in part on the transformed dequantized block; and performing control to display the displayable image via a display.

[0155] According to one embodiment, a non-transitory computer-readable storage medium stores instructions that, when executed, cause a system or device including one or more processors to: receive an entropy-coded bitstream comprising compressed video data; generate one or more dequantized blocks through a process comprising: parsing the bitstream, decoding the entropy-coded bitstream, and dequantizing each of the one or more blocks; determining whether at least one of the height and width of the one or more dequantized blocks is greater than or equal to a predetermined threshold; and, in response to the height and width of the one or more dequantized blocks being greater than or equal to the predetermined threshold, perform transform encoding on the dequantized blocks using a tuned line graph transform (LGT) core to perform direct matrix multiplication on each of the horizontal and vertical dimensions of the one or more dequantized blocks.

[0156] The instructions may be further configured to cause a system or device including one or more processors to: generate a displayable image based at least in part on the transformed dequantized block; and perform control to display the displayable image via a display.

[0157] According to one embodiment, an apparatus may include: at least one memory configured to store computer program code; and at least one processor configured to access the at least one memory and operate according to the computer program code, the computer program code including: receiving code configured to cause the at least one processor to receive an entropy-coded bitstream comprising compressed video data; generating code configured to cause the at least one processor to generate one or more dequantized blocks through a process including: parsing the bitstream, decoding the entropy-coded bitstream, and dequantizing each of the one or more blocks; determining code configured to cause the at least one processor to determine whether at least one of the height and width of the one or more dequantized blocks is greater than or equal to a predetermined threshold; and transform encoding code configured to cause the at least one processor to: in response to the height and width of the one or more dequantized blocks being greater than or equal to the predetermined threshold, transform-encode the dequantized blocks using a tuned line graph transform (LGT) core to perform direct matrix multiplication on each of the horizontal and vertical dimensions of the one or more dequantized blocks.

[0158] The apparatus may further include a display, wherein the computer program code may further include display code, the display code being configured to cause at least one processor to: generate a displayable image based at least in part on the transformed dequantized block; and perform control to display the displayable image via the display.

[0159] 8-bit transform core

[0160] 4p LGT core The 4p LGT kernel can be any 4×4 matrix from the following set: [

[0162] 41,78,105,119,

[0163] 105,105,-0,-105,

[0164] 119,-41,-105,78,

[0165] 78,-119,105,-41,

[0166] ], [

[0168] 40,77,105,119,

[0169] 103,106,2,-105,

[0170] 119,-38,-105,77,

[0171] 80,-119,103,-41,

[0172] ], [

[0174] 39,77,105,120,

[0175] 101,108,3,-105,

[0176] 120,-35,-106,76,

[0177] 82,-119,102,-40,

[0178] ], [

[0180] 38,76,105,120,

[0181] 99,109,5,-105,

[0182] 120,-32,-107,76,

[0183] 84,-118,101,-39,

[0184] ], [

[0186] 37,76,105,121,

[0187] 97,111,7,-105,

[0188] 121,-28,-108,75,

[0189] 86,-118,100,-39,

[0190] ], [

[0192] 36,75,105,121,

[0193] 96,112,8,-105,

[0194] 121,-25,-109,75,

[0195] 88,-118,98,-38,

[0196] ], [

[0198] 35,75,105,122,

[0199] 94,113,9,-105,

[0200] 121,-22,-110,75,

[0201] 91,-117,97,-37,

[0202] ], [

[0204] 34,75,106,122,

[0205] 92,115,11,-105,

[0206] 121,-18,-111,74,

[0207] 93,-117,95,-37,

[0208] ], [

[0210] 33,74,106,122,

[0211] 90,116,12,-105,

[0212] 120,-15,-112,74,

[0213] 95,-117,94,-36,

[0214] ], [

[0216] 32,74,106,123,

[0217] 88,117,13,-105,

[0218] 120,-11,-114,73,

[0219] 98,-116,92,-35,

[0220] ], [

[0222] 31,74,106,123,

[0223] 87,118,15,-105,

[0224] 119,-8,-115,73,

[0225] 100,-116,91,-34,

[0226] ], [

[0228] 30,73,106,123,

[0229] 85,119,16,-105,

[0230] 118,-5,-116,73,

[0231] 103,-115,89,-33,

[0232] ], [

[0234] 30,73,106,124,

[0235] 83,120,17,-106,

[0236] 118,-1,-117,73,

[0237] 105,-114,87,-32,

[0238] ], [

[0240] 29,73,106,124,

[0241] 82,121,18,-106,

[0242] 117,2,-118,72,

[0243] 108,-113,85,-32,

[0244] ], [

[0246] 28,73,106,124,

[0247] 80,122,19,-106,

[0248] 116,6,-119,72,

[0249] 110,-113,83,-31,

[0250] ], [

[0252] 28,72,106,124,

[0253] 79,122,20,-106,

[0254] 114,9,-120,72,

[0255] 113,-112,81,-30,

[0256] ], [

[0258] 27,72,106,125,

[0259] 77,123,21,-106,

[0260] 113,12,-121,72,

[0261] 116,-111,79,-29,

[0262] ], [

[0264] 27,72,106,125,

[0265] 75,124,22,-106,

[0266] 111,15,-122,72,

[0267] 118,-110,77,-28,

[0268] ], [

[0270] 26,72,106,125,

[0271] 74,124,23,-106,

[0272] 110,19,-124,71,

[0273] 121,-109,75,-27,

[0274] ], [

[0276] 25,71,106,125,

[0277] 73,125,24,-106,

[0278] 108,22,-125,71,

[0279] 123, -108, 73, -26,

[0280] ], [

[0282] 25,71,106,126,

[0283] 71,126,25,-106,

[0284] 106,25,-126,71,

[0285] 126, -106, 71, -25 ]

[0287] 8p LGT:

[0288] An 8p LGT core can be any 8×8 matrix from the following set: [

[0290] 27,48,67,85,99,110,118,122,

[0291] 76,115,121,93,38,-27,-85,-118,

[0292] 109,113,31,-75,-123,-78,26,111,

[0293] 123,48,-95,-105,33,124,41,-100,

[0294] 120, -41, -115, 57, 107, -72, -97, 85,

[0295] 103, -108, -10, 117, -90, -39, 124, -67,

[0296] 74, -123, 105, -31, -61, 119, -113, 46,

[0297] 39, -80, 110, -124, 121, -101, 67, -23,

[0298] ], [

[0300] 26,47,67,84,99,111,118,122,

[0301] 73,114,122,95,40,-26,-84,-119,

[0302] 106,115,35,-72,-123,-80,25,111,

[0303] 123,53,-92,-106,30,124,42,-100,

[0304] 121,-36,-116,54,108,-70,-97,85,

[0305] 105,-106,-13,117,-89,-41,124,-66,

[0306] 76,-123,104,-29,-62,119,-113,46,

[0307] 40,-81,110,-124,121,-101,67,-23,

[0308] ], [

[0310] 25,46,66,84,99,111,119,123,

[0311] 70,113,122,96,42,-25,-84,-119,

[0312] 104,117,38,-70,-124,-81,25,111,

[0313] 122,57,-89,-108,28,124,43,-99,

[0314] 122,-32,-117,51,109,-68,-98,84,

[0315] 107,-104,-16,118,-87,-42,124,-66,

[0316] 79,-123,103,-27,-63,119,-112,45,

[0317] 42,-82,111,-124,121,-100,66,-23,

[0318] ], [

[0320] 24,46,66,84,99,111,119,123,

[0321] 68,112,123,98,43,-24,-84,-119,

[0322] 102,118,42,-68,-124,-82,24,111,

[0323] 121,61,-86,-110,25,124,44,-99,

[0324] 123,-27,-118,48,110,-67,-98,84,

[0325] 109,-102,-19,119,-85,-43,124,-66,

[0326] 81,-124,101,-25,-64,119,-112,45,

[0327] 43,-83,111,-124,120,-100,66,-23,

[0328] ], [

[0330] 23,45,65,84,99,111,119,124,

[0331] 65,111,124,99,45,-23,-84,-119,

[0332] 99,119,45,-65,-124,-84,23,111,

[0333] 119,65,-84,-111,23,124,45,-99,

[0334] 124,-23,-119,45,111,-65,-99,84,

[0335] 111,-99,-23,119,-84,-45,124,-65,

[0336] 84,-124,99,-23,-65,119,-111,45,

[0337] 45,-84,111,-124,119,-99,65,-23,

[0338] ], [

[0340] 22,44,65,83,99,111,120,124,

[0341] 63,110,124,100,46,-22,-83,-120,

[0342] 97,121,48,-63,-124,-85,22,111,

[0343] 118,69,-81,-113,20,124,46,-99,

[0344] 124,-18,-120,42,112,-64,-100,83,

[0345] 113,-96,-26,120,-82,-46,124,-65,

[0346] 86,-124,97,-21,-67,120,-111,45,

[0347] 47,-85,112,-123,119,-98,65,-23,

[0348] ], [

[0350] 21,44,65,83,99,111,120,124,

[0351] 61,110,125,101,47,-21,-83,-120,

[0352] 94,122,51,-61,-124,-86,21,111,

[0353] 116,73,-78,-114,18,124,47,-99,

[0354] 124,-13,-121,39,113,-62,-100,83,

[0355] 115,-93,-30,121,-80,-48,124,-65,

[0356] 89,-124,95,-18,-68,120,-110,44,

[0357] 49,-86,112,-123,118,-98,64,-22,

[0358] ], [

[0360] 20,43,64,83,99,112,120,125,

[0361] 59,109,125,103,49,-20,-83,-120,

[0362] 92,123,53,-59,-124,-87,21,111,

[0363] 115,76,-75,-115,16,123,48,-99,

[0364] 124,-9,-122,36,114,-61,-101,83,

[0365] 117,-90,-34,121,-78,-50,124,-64,

[0366] 92,-123,93,-15,-70,120,-109,44,

[0367] 51,-87,112,-123,117,-97,64,-22,

[0368] ], [

[0370] 20,42,64,83,99,112,120,125,

[0371] 57,108,125,104,50,-20,-83,-120,

[0372] 90,123,56,-58,-124,-88,20,112,

[0373] 113,80,-73,-117,14,123,49,-99,

[0374] 124,-4,-123,32,115,-59,-102,83,

[0375] 118,-87,-38,122,-76,-51,124,-64,

[0376] 95,-123,90,-13,-71,120,-109,43,

[0377] 53,-88,113,-122,117,-96,63,-22,

[0378] ], [

[0380] 19,42,64,83,99,112,121,125,

[0381] 55,107,126,104,51,-19,-83,-121,

[0382] 87,124,58,-56,-124,-89,19,112,

[0383] 111,83,-70,-118,12,123,50,-99,

[0384] 123,1,-124,29,116,-58,-103,82,

[0385] 120,-83,-42,122,-74,-53,124,-64,

[0386] 97,-123,88,-10,-73,120,-108,43,

[0387] 55,-90,113,-122,116,-95,62,-22,

[0388] ], [

[0390] 18,42,63,83,99,112,121,125,

[0391] 54,107,126,105,52,-18,-83,-121,

[0392] 85,125,60,-54,-124,-90,19,112,

[0393] 109,86,-67,-119,9,123,51,-99,

[0394] 123,6,-124,26,117,-56,-103,82,

[0395] 121,-79,-46,123,-71,-55,124,-63,

[0396] 100,-122,85,-7,-75,120,-107,43,

[0397] 58,-91,113,-122,115,-94,61,-21,

[0398] ], [

[0400] 18,41,63,82,99,112,121,126,

[0401] 52,106,126,106,52,-18,-82,-121,

[0402] 83,125,62,-53,-124,-91,18,112,

[0403] 107,89,-65,-120,7,123,51,-99,

[0404] 122,10,-125,23,118,-54,-104,82,

[0405] 122,-75,-50,123,-69,-56,124,-63,

[0406] 103,-121,82,-3,-77,120,-107,42,

[0407] 61,-93,114,-121,114,-93,60,-21,

[0408] ], [

[0410] 17,41,63,82,99,112,121,126,

[0411] 51,105,126,107,53,-17,-82,-121,

[0412] 81,126,64,-51,-123,-92,17,112,

[0413] 105,91,-62,-121,6,123,52,-99,

[0414] 121,15,-125,20,119,-53,-105,82,

[0415] 123,-71,-54,124,-67,-58,124,-63,

[0416] 106,-120,78,0,-79,120,-106,42,

[0417] 64,-95,114,-120,113,-91,60,-21,

[0418] ], [

[0420] 17,40,62,82,99,112,121,126,

[0421] 49,105,126,108,54,-17,-82,-121,

[0422] 79,126,66,-50,-123,-93,17,112,

[0423] 103,94,-60,-122,4,123,53,-99,

[0424] 120,19,-125,17,120,-51,-105,82,

[0425] 124,-67,-58,124,-65,-60,124,-62,

[0426] 109,-119,75,4,-81,121,-105,41,

[0427] 67,-96,115,-120,111,-90,58,-20,

[0428] ], [

[0430] 16,40,62,82,99,112,121,126,

[0431] 48,104,127,108,55,-16,-82,-121,

[0432] 77,126,68,-48,-123,-93,16,112,

[0433] 101,96,-57,-122,2,123,54,-99,

[0434] 118,24,-125,14,121,-50,-106,82,

[0435] 124,-62,-62,124,-62,-62,124,-62,

[0436] 112,-118,71,8,-83,121,-104,41,

[0437] 71,-98,115,-119,110,-88,57,-20,

[0438] ], [

[0440] 16,40,62,82,99,112,122,126,

[0441] 46,103,127,109,56,-16,-82,-122,

[0442] 75,127,70,-47,-123,-94,16,112,

[0443] 99,98,-55,-123,0,123,55,-99,

[0444] 117,28,-125,11,122,-49,-107,82,

[0445] 124,-57,-66,124,-60,-64,124,-62,

[0446] 115,-116,66,12,-86,121,-104,40,

[0447] 75,-100,115,-118,108,-87,56,-19,

[0448] ], [

[0450] 15,39,62,82,99,112,122,126,

[0451] 45,103,127,109,56,-15,-82,-122,

[0452] 73,127,71,-46,-123,-95,16,112,

[0453] 97,101,-53,-124,-1,123,55,-99,

[0454] 115,32,-125,8,123,-47,-108,82,

[0455] 124,-53,-70,124,-57,-66,125,-62,

[0456] 117,-114,62,17,-88,122,-103,40,

[0457] 79,-103,116,-117,106,-85,55,-19,

[0458] ], [

[0460] 15,39,62,82,99,113,122,127,

[0461] 44,102,127,110,57,-15,-82,-122,

[0462] 71,127,73,-44,-123,-95,15,112,

[0463] 95,102,-51,-124,-3,123,56,-99,

[0464] 113,36,-125,5,123,-46,-108,82,

[0465] 124,-48,-74,124,-55,-68,125,-61,

[0466] 119,-111,57,22,-91,122,-102,40,

[0467] 84,-105,116,-116,104,-83,53,-18,

[0468] ], [

[0470] 15,39,61,82,99,113,122,127,

[0471] 43,102,127,110,57,-15,-82,-122,

[0472] 69,127,74,-43,-123,-96,15,112,

[0473] 93,104,-48,-125,-5,123,57,-99,

[0474] 111,40,-125,2,124,-44,-109,81,

[0475] 123,-43,-78,124,-52,-70,125,-61,

[0476] 121,-108,51,27,-94,123,-102,39,

[0477] 89,-107,116,-114,102,-80,51,-18,

[0478] ], [

[0480] 14,38,61,82,99,113,122,127,

[0481] 42,101,127,111,58,-14,-82,-122,

[0482] 68,127,76,-42,-123,-96,14,113,

[0483] 91,106,-46,-125,-6,123,57,-99,

[0484] 109,44,-125,-0,125,-43,-110,81,

[0485] 122, -38, -82, 124, -50, -72, 126, -61,

[0486] 123, -105, 46, 32, -97, 123, -101, 39,

[0487] 94, -110, 116, -112, 99, -78, 49, -17,

[0488] ], [

[0490] 14,38,61,82,99,113,122,127,

[0491] 41,101,127,111,58,-14,-82,-122,

[0492] 66,127,77,-41,-123,-97,14,113,

[0493] 89,108,-44,-126,-7,123,58,-99,

[0494] 107,47,-124,-3,125,-42,-110,81,

[0495] 120, -32, -86, 124, -47, -74, 126, -61,

[0496] 124, -101, 39, 37, -100, 124, -101, 38,

[0497] 100, -112, 116, -110, 96, -75, 47, -16, ]

[0499] 16p LGT:

[0500] A 16p LGT kernel can be any 16×16 matrix from the following set: [

[0502] 22,33,43,53,63,72,81,88,96,102,108,113,116,119,121,122,

[0503] 61,88,108,120,123,117,103,82,54,23,-10,-43,-72,-96,-113,-122,

[0504] 92,119,122,101,59,6,-49,-94,-120,-122,-99,-56,-2,53,96,121,

[0505] 111,123,86,16,-61,-114,-122,-82,-9,67,116,120,77,3,-72,-118,

[0506] 122,104,19,-78,-125,-92,1,93,125,77,-21,-105,-122,-60,41,115,

[0507] 126,70,-52,-125,-79,41,123,88,-30,-120,-97,18,115,104,-6,-110,

[0508] 125,27,-106,-102,33,126,57,-85,-117,2,119,83,-60,-125,-30,104,

[0509] 120,-19,-127,-25,118,65,-96,-98,62,119,-21,-127,-23,119,64,-97,

[0510] 113,-61,-110,65,108,-69,-105,73,102,-77,-99,81,96,-85,-92,89,

[0511] 103,-95,-63,121,12,-126,42,108,-88,-71,118,22,-127,32,113,-80,

[0512] 91,-117,2,116,-93,-42,127,-58,-81,122,-15,-110,102,29,-125,70,

[0513] 78,-127,65,53,-125,88,25,-117,107,-3,-103,120,-32,-83,126,-59,

[0514] 64,-123,111,-36,-60,121,-113,40,56,-120,115,-44,-52,119,-117,48,

[0515] 49,-105,127,-107,52,21,-86,124,-120,77,-8,-63,114,-126,98,-37,

[0516] 33,-77,110,-126,124,-103,67,-20,-29,74,-107,125,-125,105,-70,25,

[0517] 16,-41,63,-83,100,-114,123,-127,127,-121,112,-98,80,-59,37,-12,

[0518] ], [

[0520] 20,31,42,52,62,71,80,88,96,102,108,113,117,120,122,123,

[0521] 57,85,106,119,123,119,105,84,57,25,-9,-42,-71,-96,-113,-122,

[0522] 87,117,123,104,64,10,-46,-92,-119,-122,-100,-58,-3,52,96,121,

[0523] 108,124,91,22,-56,-112,-123,-85,-13,64,115,121,78,4,-71,-118,

[0524] 120,108,26,-73,-125,-95,-4,90,125,79,-19,-104,-122,-61,40,115,

[0525] 125,75,-47,-124,-83,37,122,91,-26,-119,-98,16,115,104,-5,-110,

[0526] 126,32,-103,-105,29,125,60,-83,-118,-1,118,84,-58,-125,-30,104,

[0527] 122,-14,-126,-29,116,68,-93,-100,59,120,-19,-126,-24,118,64,-97,

[0528] 115,-57,-112,62,109,-66,-106,71,103,-76,-100,80,96,-84,-93,89,

[0529] 105,-92,-66,120,15,-126,39,109,-86,-73,117,23,-127,31,114,-80,

[0530] 93,-116,-1,117,-92,-44,127,-56,-82,121,-14,-110,102,30,-125,70,

[0531] 80,-127,63,55,-126,87,27,-118,106,-2,-103,119,-32,-84,126,-59,

[0532] 66,-123,110,-34,-61,122,-113,39,57,-120,115,-44,-53,119,-117,48,

[0533] 50,-106,127,-106,51,22,-87,124,-120,76,-8,-63,114,-126,97,-36,

[0534] 34,-78,110,-126,123,-102,66,-20,-29,74,-108,125,-124,105,-70,25,

[0535] 17,-41,63,-84,100,-114,123,-127,126,-121,111,-98,80,-59,37,-12,

[0536] ], [

[0538] 19,30,41,51,61,71,80,88,96,102,108,113,117,120,122,123,

[0539] 54,82,104,119,124,120,107,86,59,27,-7,-41,-71,-96,-114,-123,

[0540] 83,115,124,107,67,14,-42,-90,-119,-123,-102,-59,-5,51,96,121,

[0541] 104,125,95,27,-52,-110,-124,-88,-16,61,115,121,79,5,-71,-118,

[0542] 118,111,32,-68,-124,-98,-8,87,125,81,-16,-103,-122,-62,40,115,

[0543] 125,80,-41,-123,-87,32,121,93,-23,-118,-99,14,114,105,-5,-110,

[0544] 126,38,-99,-108,24,124,63,-80,-119,-4,117,85,-57,-125,-31,104,

[0545] 123,-8,-126,-34,115,72,-91,-102,57,121,-17,-126,-25,118,64,-97,

[0546] 117,-52,-114,58,111,-64,-108,69,105,-74,-101,79,97,-84,-93,88,

[0547] 107,-89,-69,119,18,-126,37,111,-84,-74,116,24,-127,30,114,-79,

[0548] 96,-115,-4,118,-90,-47,127,-54,-83,121,-13,-111,101,30,-125,70,

[0549] 82,-126,61,57,-126,85,29,-118,105,-1,-104,119,-31,-84,126,-59,

[0550] 68,-124,109,-32,-63,122,-112,38,58,-120,114,-43,-53,119,-117,48,

[0551] 52,-107,127,-106,49,23,-88,124,-119,75,-7,-64,114,-126,97,-36,

[0552] 35,-78,110,-126,123,-102,65,-19,-30,74,-108,125,-124,105,-70,24,

[0553] 18,-41,64,-84,101,-114,123,-127,126,-121,111,-97,80,-59,36,-12,

[0554] ], [

[0556] 18,29,40,50,61,70,79,88,96,102,108,114,118,121,123,124,

[0557] 51,80,103,118,124,121,108,88,60,28,-6,-40,-70,-96,-114,-123,

[0558] 79,114,125,109,71,18,-39,-88,-118,-123,-103,-61,-6,50,96,121,

[0559] 101,125,99,32,-47,-108,-124,-90,-19,59,114,122,81,6,-70,-118,

[0560] 115,114,37,-64,-123,-101,-12,85,125,83,-14,-102,-123,-62,39,115,

[0561] 123,85,-36,-122,-90,28,119,96,-20,-117,-101,12,113,105,-4,-110,

[0562] 126,43,-96,-110,19,124,66,-77,-120,-6,116,87,-56,-125,-32,103,

[0563] 124,-3,-125,-38,113,75,-88,-104,54,121,-15,-126,-26,118,65,-96,

[0564] 118,-48,-116,54,113,-60,-109,66,106,-72,-102,78,98,-83,-93,88,

[0565] 109,-86,-72,117,21,-126,34,112,-83,-76,116,25,-127,30,114,-79,

[0566] 98,-113,-8,119,-88,-49,127,-52,-85,120,-12,-111,100,31,-125,69,

[0567] 85,-126,58,59,-126,84,31,-119,104,0,-105,119,-30,-84,126,-59,

[0568] 70,-124,108,-30,-64,122,-111,36,59,-121,114,-42,-53,119,-116,48,

[0569] 53,-108,127,-105,48,24,-89,124,-119,75,-6,-64,114,-126,97,-36,

[0570] 36,-79,111,-126,123,-101,65,-19,-30,75,-108,125,-124,105,-70,24,

[0571] 18,-42,64,-84,101,-114,123,-127,126,-121,111,-97,80,-59,36,-12,

[0572] ], [

[0574] 16,28,39,50,60,70,79,88,96,102,109,114,118,121,123,124,

[0575] 48,78,101,117,124,122,110,89,62,30,-5,-39,-70,-96,-114,-123,

[0576] 75,112,125,111,74,21,-36,-86,-117,-124,-104,-62,-7,50,96,121,

[0577] 97,125,102,37,-43,-106,-125,-92,-22,57,113,122,82,7,-70,-119,

[0578] 113,116,43,-59,-122,-103,-16,82,125,85,-11,-101,-123,-63,38,115,

[0579] 122,89,-30,-120,-94,23,118,98,-17,-115,-102,10,113,106,-3,-109,

[0580] 126,49,-92,-112,15,123,70,-75,-121,-9,115,88,-54,-125,-32,103,

[0581] 125,3,-124,-42,111,78,-86,-105,52,122,-13,-126,-28,117,65,-96,

[0582] 120,-43,-117,50,114,-57,-111,64,107,-70,-103,76,98,-82,-93,88,

[0583] 112,-83,-75,116,24,-126,31,113,-81,-77,115,27,-127,29,114,-79,

[0584] 101,-111,-11,120,-85,-51,127,-50,-86,120,-10,-112,100,31,-125,69,

[0585] 87,-125,56,62,-126,82,33,-119,103,2,-105,118,-29,-85,126,-59,

[0586] 72,-124,106,-28,-66,123,-110,35,60,-121,113,-41,-54,119,-116,48,

[0587] 55,-109,127,-104,47,26,-89,124,-118,74,-6,-65,114,-126,97,-36,

[0588] 37,-80,111,-126,122,-100,64,-18,-31,75,-108,125,-124,104,-69,24,

[0589] 19,-43,65,-84,101,-114,123,-127,126,-121,111,-97,79,-59,36,-12,

[0590] ], [

[0592] 15,27,38,49,60,70,79,88,95,103,109,114,118,121,124,125,

[0593] 45,76,100,117,124,122,111,90,63,31,-4,-38,-70,-96,-114,-124,

[0594] 72,110,125,113,77,24,-34,-84,-117,-124,-105,-63,-8,49,96,122,

[0595] 94,125,105,41,-40,-104,-125,-95,-25,55,112,123,83,8,-70,-119,

[0596] 110,118,48,-55,-121,-106,-20,79,125,87,-9,-100,-123,-64,38,115,

[0597] 120,93,-25,-119,-97,19,117,100,-14,-114,-103,8,112,107,-3,-109,

[0598] 125,54,-88,-115,10,121,73,-72,-122,-11,114,89,-53,-125,-33,103,

[0599] 126,8,-123,-47,108,81,-83,-107,49,122,-11,-126,-29,117,66,-96,

[0600] 122,-38,-119,46,116,-54,-112,61,108,-68,-104,75,99,-82,-94,88,

[0601] 114,-79,-79,114,28,-126,28,114,-79,-79,114,28,-126,28,114,-79,

[0602] 103,-109,-15,121,-83,-54,127,-48,-88,119,-9,-112,99,32,-125,69,

[0603] 90,-125,53,64,-126,80,35,-120,102,3,-106,118,-29,-85,126,-59,

[0604] 74,-125,105,-26,-68,123,-109,33,61,-121,113,-41,-55,119,-116,48,

[0605] 57,-110,127,-103,45,27,-90,124,-118,73,-5,-65,114,-126,97,-36,

[0606] 39,-81,112,-126,122,-100,63,-17,-31,75,-108,125,-124,104,-69,24,

[0607] 19,-43,65,-85,101,-114,123,-127,126,-120,111,-97,79,-59,36,-12,

[0608] ], [

[0610] 15,26,38,49,59,69,79,87,95,103,109,114,118,122,124,125,

[0611] 43,74,99,116,125,123,112,92,64,32,-3,-38,-69,-95,-114,-124,

[0612] 69,109,125,115,79,27,-31,-83,-116,-125,-106,-65,-9,48,95,122,

[0613] 90,125,107,45,-36,-102,-125,-96,-27,53,112,123,84,9,-69,-119,

[0614] 107,120,53,-51,-120,-108,-23,77,125,89,-7,-99,-124,-65,37,115,

[0615] 118,97,-19,-117,-100,15,115,102,-11,-113,-105,6,111,107,-2,-109,

[0616] 125,59,-84,-117,5,120,76,-69,-123,-14,113,91,-52,-126,-33,103,

[0617] 126,14,-122,-51,106,84,-80,-109,46,123,-8,-126,-30,116,66,-96,

[0618] 123,-33,-120,42,117,-50,-114,58,110,-66,-105,74,100,-81,-94,88,

[0619] 116,-75,-82,112,32,-126,25,115,-77,-80,113,29,-126,27,114,-79,

[0620] 105,-107,-19,122,-80,-57,126,-46,-89,118,-7,-112,98,33,-125,69,

[0621] 92,-124,50,67,-126,78,37,-120,101,5,-106,117,-28,-85,125,-58,

[0622] 76,-125,103,-24,-70,124,-108,32,62,-121,112,-40,-55,119,-116,47,

[0623] 59,-110,126,-101,43,28,-91,124,-117,72,-4,-66,114,-126,96,-36,

[0624] 40,-82,112,-126,121,-99,62,-16,-32,76,-108,125,-123,104,-69,24,

[0625] 20,-44,66,-85,102,-114,123,-126,126,-120,110,-96,79,-59,36,-12,

[0626] ], [

[0628] 14,26,37,48,59,69,78,87,95,103,109,114,119,122,124,125,

[0629] 41,72,98,116,125,124,113,93,65,33,-2,-37,-69,-95,-114,-124,

[0630] 66,107,125,116,82,29,-29,-81,-116,-125,-107,-66,-10,48,95,122,

[0631] 87,124,109,49,-33,-100,-126,-98,-30,51,111,124,85,10,-69,-119,

[0632] 104,122,57,-47,-118,-110,-27,75,125,91,-5,-98,-124,-66,37,115,

[0633] 116,100,-14,-115,-102,11,114,104,-8,-112,-106,5,111,108,-2,-109,

[0634] 124,64,-80,-118,1,119,79,-66,-123,-17,112,92,-50,-126,-34,103,

[0635] 126,19,-120,-56,103,87,-77,-110,44,123,-6,-125,-32,116,67,-96,

[0636] 124,-28,-122,38,119,-47,-115,56,111,-64,-106,72,100,-80,-94,87,

[0637] 118,-71,-85,110,35,-126,22,116,-75,-82,112,31,-126,27,114,-78,

[0638] 108,-104,-23,123,-78,-59,126,-44,-90,117,-5,-113,98,33,-125,69,

[0639] 95,-123,47,69,-126,76,39,-121,100,6,-107,117,-27,-86,125,-58,

[0640] 79,-125,102,-21,-72,124,-107,30,64,-122,111,-39,-56,119,-115,47,

[0641] 61,-111,126,-100,42,30,-92,124,-117,71,-3,-66,114,-125,96,-36,

[0642] 42,-83,113,-126,121,-98,61,-15,-33,76,-109,125,-123,104,-69,24,

[0643] 21,-44,66,-86,102,-114,123,-126,125,-120,110,-96,79,-58,36,-12,

[0644] ], [

[0646] 13,25,36,48,58,69,78,87,95,103,109,114,119,122,124,125,

[0647] 39,71,97,115,125,124,113,94,66,34,-1,-36,-69,-95,-114,-124,

[0648] 63,105,125,117,84,32,-27,-80,-115,-125,-108,-67,-11,48,95,122,

[0649] 84,124,111,52,-30,-98,-126,-100,-32,50,110,124,85,11,-69,-119,

[0650] 101,123,61,-43,-117,-112,-30,72,125,92,-3,-97,-124,-67,36,115,

[0651] 114,104,-9,-113,-105,7,112,106,-5,-111,-107,3,110,108,-1,-109,

[0652] 122,69,-76,-120,-3,118,82,-63,-124,-19,111,93,-49,-126,-34,103,

[0653] 126,25,-119,-60,101,90,-74,-112,41,124,-4,-125,-33,115,67,-95,

[0654] 125,-23,-123,33,120,-43,-117,53,112,-62,-107,71,101,-79,-95,87,

[0655] 119,-67,-89,108,39,-126,19,117,-73,-84,111,32,-126,26,114,-78,

[0656] 110,-101,-27,123,-75,-62,126,-41,-92,117,-4,-114,97,34,-125,69,

[0657] 97,-122,43,72,-126,73,42,-121,98,8,-108,116,-26,-86,125,-58,

[0658] 81,-126,100,-18,-74,124,-106,28,65,-122,111,-38,-56,119,-115,47,

[0659] 63,-113,126,-99,40,32,-93,125,-116,70,-2,-67,114,-125,96,-36,

[0660] 43,-84,113,-126,120,-97,60,-14,-34,77,-109,125,-123,103,-69,24,

[0661] 22,-45,67,-86,102,-114,123,-126,125,-120,110,-96,78,-58,36,-12,

[0662] ], [

[0664] 13,24,36,47,58,68,78,87,95,103,109,115,119,122,125,126,

[0665] 37,69,96,115,125,124,114,94,67,35,-1,-36,-68,-95,-115,-125,

[0666] 60,104,125,118,85,34,-26,-79,-115,-126,-109,-67,-11,47,95,122,

[0667] 81,123,113,55,-27,-97,-126,-101,-34,48,110,124,86,11,-68,-119,

[0668] 98,124,65,-39,-116,-113,-33,70,125,94,-2,-96,-124,-67,36,115,

[0669] 112,107,-5,-111,-107,4,111,108,-3,-110,-108,2,110,109,-1,-109,

[0670] 121,74,-72,-121,-8,116,84,-60,-124,-22,110,94,-48,-126,-35,103,

[0671] 125,30,-117,-64,98,93,-71,-113,38,124,-2,-125,-34,115,68,-95,

[0672] 126,-17,-124,29,121,-39,-118,50,113,-60,-108,70,102,-79,-95,87,

[0673] 121,-62,-92,105,43,-125,15,118,-70,-85,110,34,-126,25,114,-78,

[0674] 112,-98,-31,124,-71,-65,125,-38,-94,116,-2,-114,96,35,-125,68,

[0675] 100,-121,39,75,-126,71,44,-122,97,10,-108,115,-25,-87,125,-58,

[0676] 84,-126,98,-15,-76,124,-104,26,67,-122,110,-37,-57,119,-115,47,

[0677] 66,-114,125,-97,38,34,-94,125,-115,69,-1,-67,115,-125,96,-36,

[0678] 45,-86,114,-126,120,-96,59,-13,-35,77,-109,125,-123,103,-68,24,

[0679] 23,-46,67,-86,102,-115,123,-126,125,-119,110,-96,78,-58,36,-12,

[0680] ], [

[0682] 12,24,36,47,58,68,78,87,95,103,109,115,119,122,125,126,

[0683] 36,68,95,115,125,125,115,95,68,36,0,-36,-68,-95,-115,-125,

[0684] 58,103,125,119,87,36,-24,-78,-115,-126,-109,-68,-12,47,95,122,

[0685] 78,122,115,58,-24,-95,-126,-103,-36,47,109,125,87,12,-68,-119,

[0686] 95,125,68,-36,-115,-115,-36,68,125,95,-0,-95,-125,-68,36,115,

[0687] 109,109,0,-109,-109,-0,109,109,0,-109,-109,-0,109,109,0,-109,

[0688] 119,78,-68,-122,-12,115,87,-58,-125,-24,109,95,-47,-126,-36,103,

[0689] 125,36,-115,-68,95,95,-68,-115,36,125,-0,-125,-36,115,68,-95,

[0690] 126,-12,-125,24,122,-36,-119,47,115,-58,-109,68,103,-78,-95,87,

[0691] 122,-58,-95,103,47,-125,12,119,-68,-87,109,36,-126,24,115,-78,

[0692] 115,-95,-36,125,-68,-68,125,-36,-95,115,-0,-115,95,36,-125,68,

[0693] 103,-119,36,78,-126,68,47,-122,95,12,-109,115,-24,-87,125,-58,

[0694] 87,-126,95,-12,-78,125,-103,24,68,-122,109,-36,-58,119,-115,47,

[0695] 68,-115,125,-95,36,36,-95,125,-115,68,0,-68,115,-125,95,-36,

[0696] 47,-87,115,-126,119,-95,58,-12,-36,78,-109,125,-122,103,-68,24,

[0697] 24,-47,68,-87,103,-115,122,-126,125,-119,109,-95,78,-58,36,-12,

[0698] ], [

[0700] 11,23,35,47,57,68,78,87,95,103,109,115,119,123,125,126,

[0701] 34,67,94,114,125,125,115,96,69,36,1,-35,-68,-95,-115,-125,

[0702] 56,101,125,120,88,37,-22,-77,-114,-126,-110,-69,-13,47,95,123,

[0703] 75,122,116,60,-21,-94,-126,-104,-37,46,109,125,88,13,-68,-119,

[0704] 92,125,71,-32,-113,-116,-38,66,125,97,2,-95,-125,-69,35,115,

[0705] 107,112,4,-107,-111,-3,108,111,2,-108,-110,-1,109,110,0,-109,

[0706] 117,82,-64,-124,-16,113,89,-55,-125,-26,108,96,-46,-126,-36,103,

[0707] 124,41,-112,-72,92,98,-65,-116,33,125,2,-125,-37,114,69,-95,

[0708] 126,-6,-125,19,123,-32,-120,44,116,-56,-110,67,103,-77,-96,87,

[0709] 124,-53,-98,100,51,-124,8,120,-66,-89,108,37,-126,23,115,-78,

[0710] 117,-92,-40,125,-65,-71,124,-33,-97,114,2,-115,94,36,-125,68,

[0711] 105,-117,31,81,-126,65,50,-123,94,14,-110,114,-23,-87,125,-58,

[0712] 90,-126,93,-9,-80,125,-101,22,70,-123,108,-34,-59,119,-114,47,

[0713] 71,-116,124,-93,33,38,-96,125,-114,67,1,-69,115,-125,95,-35,

[0714] 49,-88,115,-126,118,-94,56,-11,-37,79,-109,125,-122,102,-68,24,

[0715] 25,-48,69,-88,103,-115,122,-126,124,-119,109,-95,78,-57,35,-12,

[0716] ], [

[0718] 11,23,35,46,57,68,78,87,95,103,109,115,119,123,125,126,

[0719] 33,66,94,114,125,125,116,97,70,37,1,-35,-68,-95,-115,-125,

[0720] 53,100,124,121,90,39,-21,-76,-114,-126,-110,-70,-13,46,95,123,

[0721] 73,121,117,63,-19,-92,-126,-105,-39,44,108,125,88,13,-68,-119,

[0722] 90,126,74,-29,-112,-117,-41,64,124,98,3,-94,-125,-69,35,115,

[0723] 104,114,9,-105,-113,-7,106,112,5,-107,-111,-3,108,110,1,-109,

[0724] 115,86,-60,-124,-20,112,92,-52,-126,-28,107,97,-45,-126,-37,103,

[0725] 123,46,-110,-76,90,100,-62,-117,30,125,4,-124,-38,114,69,-95,

[0726] 126,-1,-126,14,124,-28,-121,41,117,-53,-111,65,104,-76,-96,87,

[0727] 125,-48,-102,97,55,-123,5,121,-63,-90,107,39,-126,22,115,-78,

[0728] 119,-88,-45,126,-61,-74,123,-30,-99,113,4,-116,93,37,-125,68,

[0729] 108,-115,27,84,-125,62,52,-123,92,16,-111,113,-22,-88,124,-57,

[0730] 93,-126,90,-5,-83,125,-99,19,72,-123,107,-33,-59,119,-114,46,

[0731] 74,-117,123,-91,30,40,-98,125,-113,66,2,-70,115,-124,95,-35,

[0732] 51,-90,116,-126,118,-93,55,-9,-38,79,-110,125,-122,102,-68,24,

[0733] 26,-49,70,-88,103,-115,122,-126,124,-118,108,-94,77,-57,35,-12,

[0734] ], [

[0736] 11,23,34,46,57,68,78,87,95,103,109,115,119,123,125,126,

[0737] 31,65,93,114,125,126,116,97,70,37,2,-34,-68,-95,-115,-125,

[0738] 51,99,124,121,91,40,-20,-75,-114,-126,-111,-70,-14,46,95,123,

[0739] 70,120,118,65,-17,-91,-126,-106,-40,43,108,126,89,14,-68,-119,

[0740] 87,126,77,-26,-111,-118,-43,63,124,99,4,-93,-126,-70,34,115,

[0741] 102,116,12,-103,-114,-10,105,113,7,-106,-112,-4,108,111,1,-109,

[0742] 113,89,-56,-125,-24,110,94,-50,-126,-30,106,98,-44,-126,-37,103,

[0743] 121,51,-108,-79,87,103,-59,-118,28,126,6,-124,-39,114,70,-95,

[0744] 126,5,-126,10,125,-24,-122,38,118,-51,-112,64,105,-76,-96,87,

[0745] 125,-43,-105,94,59,-122,1,122,-61,-92,106,40,-126,21,115,-77,

[0746] 121,-84,-49,126,-57,-77,123,-26,-100,111,6,-116,93,38,-125,68,

[0747] 111,-113,22,87,-125,59,55,-124,90,18,-112,112,-20,-89,124,-57,

[0748] 96,-125,87,-1,-85,125,-97,16,73,-123,106,-32,-60,119,-114,46,

[0749] 77,-118,123,-89,28,42,-99,125,-112,64,4,-70,115,-124,94,-35,

[0750] 54,-92,117,-125,117,-91,53,-8,-39,80,-110,124,-121,101,-67,24,

[0751] 28,-50,71,-89,104,-115,122,-125,124,-118,108,-94,77,-57,35,-12,

[0752] ], [

[0754] 10,22,34,46,57,67,77,87,95,103,109,115,120,123,125,127,

[0755] 30,64,92,113,125,126,117,98,71,38,2,-34,-67,-95,-115,-125,

[0756] 50,98,124,122,92,42,-18,-74,-113,-127,-111,-71,-14,46,95,123,

[0757] 68,120,119,67,-15,-90,-126,-107,-42,42,107,126,89,14,-67,-120,

[0758] 84,127,80,-23,-110,-119,-45,61,124,100,6,-93,-126,-71,34,115,

[0759] 99,117,16,-101,-116,-13,103,114,9,-105,-113,-5,107,111,2,-109,

[0760] 111,93,-52,-126,-27,108,96,-47,-126,-32,106,99,-43,-126,-38,103,

[0761] 120,55,-105,-83,84,105,-56,-119,25,126,8,-124,-40,113,70,-95,

[0762] 125,10,-126,5,126,-20,-123,35,119,-49,-113,62,106,-75,-97,87,

[0763] 126,-37,-107,90,63,-121,-3,123,-58,-94,105,42,-125,20,116,-77,

[0764] 122,-80,-54,126,-53,-81,122,-23,-102,110,8,-117,92,39,-125,67,

[0765] 113,-111,17,90,-124,55,59,-124,88,21,-112,112,-19,-89,124,-57,

[0766] 99,-125,83,3,-88,125,-95,14,75,-124,105,-30,-61,120,-113,46,

[0767] 80,-119,122,-86,24,45,-101,125,-111,62,5,-71,115,-124,94,-35,

[0768] 56,-93,117,-125,116,-90,52,-6,-40,81,-110,124,-121,101,-67,23,

[0769] 29,-51,72,-89,104,-115,122,-125,123,-117,107,-94,76,-57,35,-12,

[0770] ], [

[0772] 10,22,34,45,57,67,77,87,95,103,109,115,120,123,125,127,

[0773] 29,63,92,113,125,126,117,98,71,38,2,-34,-67,-95,-115,-125,

[0774] 48,97,124,122,93,43,-17,-73,-113,-127,-112,-71,-15,45,95,123,

[0775] 66,119,120,69,-13,-89,-126,-108,-43,41,107,126,90,15,-67,-120,

[0776] 82,127,82,-20,-109,-120,-47,59,124,101,7,-92,-126,-71,34,115,

[0777] 97,119,20,-99,-117,-15,102,115,11,-105,-113,-7,107,111,2,-109,

[0778] 109,96,-49,-126,-31,107,98,-45,-126,-34,105,100,-42,-126,-38,103,

[0779] 118,60,-103,-86,81,107,-53,-120,22,126,10,-124,-42,113,71,-95,

[0780] 124,15,-126,0,126,-16,-124,32,120,-47,-114,61,107,-74,-97,86,

[0781] 126,-32,-110,87,67,-120,-6,123,-56,-96,103,44,-125,19,116,-77,

[0782] 123,-75,-59,126,-49,-84,121,-20,-104,109,10,-118,91,40,-125,67,

[0783] 116,-108,12,94,-123,52,62,-125,86,23,-113,111,-18,-90,124,-57,

[0784] 102,-124,80,8,-91,125,-93,11,77,-124,104,-29,-62,120,-113,46,

[0785] 83,-120,120,-83,21,48,-102,125,-110,61,7,-72,116,-123,93,-35,

[0786] 59,-95,118,-125,114,-88,50,-4,-42,82,-111,124,-121,100,-66,23,

[0787] 31,-53,73,-90,105,-115,122,-125,123,-117,107,-93,76,-56,35,-12,

[0788] ], [

[0790] 9,22,34,45,56,67,77,87,95,103,109,115,120,123,126,127,

[0791] 28,62,91,113,125,126,117,99,72,39,3,-34,-67,-95,-115,-126,

[0792] 46,96,123,123,94,44,-16,-73,-113,-127,-112,-72,-15,45,95,123,

[0793] 64,118,121,71,-11,-87,-126,-109,-44,40,107,126,90,15,-67,-120,

[0794] 80,127,84,-18,-107,-121,-49,58,123,102,8,-91,-126,-72,34,115,

[0795] 94,120,23,-97,-118,-18,101,116,13,-104,-114,-8,107,112,3,-109,

[0796] 106,98,-45,-126,-34,105,100,-43,-127,-36,104,101,-41,-127,-38,103,

[0797] 116,64,-100,-89,78,109,-50,-121,20,126,12,-123,-43,113,71,-95,

[0798] 123,21,-126,-4,126,-12,-125,29,121,-44,-115,59,107,-74,-98,86,

[0799] 126,-26,-113,83,70,-119,-10,124,-53,-97,102,45,-125,18,116,-77,

[0800] 125,-70,-64,125,-44,-87,120,-16,-105,107,13,-118,90,41,-125,67,

[0801] 118,-105,7,97,-122,48,65,-125,83,26,-114,110,-16,-91,124,-57,

[0802] 106,-123,76,12,-94,125,-90,7,79,-124,103,-27,-63,120,-113,46,

[0803] 87,-121,119,-80,17,51,-104,125,-108,59,9,-73,116,-123,93,-34,

[0804] 62,-97,119,-124,113,-86,47,-2,-43,83,-111,124,-120,100,-66,23,

[0805] 32,-54,74,-91,105,-116,122,-125,123,-116,106,-92,75,-56,34,-12,

[0806] ], [

[0808] 9,21,33,45,56,67,77,86,95,103,109,115,120,123,126,127,

[0809] 27,62,91,112,125,126,118,99,72,39,3,-33,-67,-95,-115,-126,

[0810] 45,95,123,123,95,45,-15,-72,-112,-127,-113,-72,-15,45,95,123,

[0811] 62,118,122,72,-9,-86,-126,-109,-45,39,106,126,91,15,-67,-120,

[0812] 77,127,86,-16,-106,-122,-51,56,123,103,9,-91,-126,-72,33,115,

[0813] 92,121,26,-96,-119,-21,99,117,15,-103,-115,-9,106,112,3,-109,

[0814] 104,101,-41,-127,-37,104,102,-41,-127,-38,103,102,-40,-127,-39,103,

[0815] 114,68,-98,-92,75,111,-48,-122,18,127,14,-123,-44,112,72,-95,

[0816] 122,26,-125,-9,126,-8,-125,26,122,-42,-116,58,108,-73,-98,86,

[0817] 126,-21,-115,80,74,-118,-14,125,-50,-99,101,47,-125,17,116,-77,

[0818] 125,-65,-69,125,-40,-90,118,-13,-107,106,15,-119,89,42,-125,67,

[0819] 120,-101,1,100,-121,44,69,-125,81,29,-115,109,-15,-91,124,-56,

[0820] 109,-122,71,17,-97,125,-88,4,82,-124,101,-25,-65,120,-112,45,

[0821] 91,-122,117,-77,13,54,-105,125,-107,57,11,-75,116,-123,92,-34,

[0822] 66,-99,120,-124,111,-84,45,0,-45,84,-111,124,-120,99,-65,23,

[0823] 35,-56,75,-92,106,-116,122,-124,122,-116,105,-92,75,-55,34,-11,

[0824] ], [

[0826] 9,21,33,45,56,67,77,86,95,103,110,115,120,123,126,127,

[0827] 26,61,90,112,125,127,118,100,73,40,4,-33,-67,-95,-115,-126,

[0828] 43,94,123,124,96,46,-14,-71,-112,-127,-113,-73,-16,45,95,123,

[0829] 60,117,122,74,-7,-85,-126,-110,-46,38,106,127,91,16,-67,-120,

[0830] 75,127,88,-13,-105,-122,-52,55,123,103,10,-90,-126,-72,33,115,

[0831] 89,122,29,-94,-120,-23,98,118,16,-102,-115,-10,106,113,3,-109,

[0832] 102,103,-38,-127,-40,102,103,-38,-127,-40,102,103,-39,-127,-39,103,

[0833] 112,72,-95,-95,72,112,-45,-123,15,127,15,-123,-45,112,72,-95,

[0834] 120,31,-125,-13,126,-5,-126,23,123,-40,-117,57,109,-72,-98,86,

[0835] 125,-15,-117,76,78,-116,-18,125,-48,-101,100,49,-125,16,117,-77,

[0836] 126,-60,-73,124,-35,-93,117,-9,-109,105,17,-120,88,43,-125,67,

[0837] 122,-97,-5,103,-119,39,72,-126,79,31,-116,108,-13,-92,124,-56,

[0838] 112,-120,66,23,-100,125,-85,0,84,-125,100,-23,-66,120,-112,45,

[0839] 95,-123,115,-73,9,58,-107,125,-105,54,13,-76,117,-122,92,-34,

[0840] 69,-102,120,-123,110,-81,42,3,-47,85,-112,124,-119,99,-65,23,

[0841] 37,-58,77,-93,106,-116,122,-124,121,-115,105,-91,74,-55,34,-11,

[0842] ], [

[0844] 9,21,33,45,56,67,77,86,95,103,110,115,120,124,126,127,

[0845] 25,60,90,112,125,127,118,100,73,40,4,-33,-67,-95,-115,-126,

[0846] 42,93,123,124,97,47,-13,-71,-112,-127,-113,-73,-16,45,95,123,

[0847] 58,116,123,75,-6,-84,-125,-111,-47,38,106,127,92,16,-67,-120,

[0848] 73,127,90,-11,-104,-123,-54,54,123,104,11,-90,-127,-73,33,115,

[0849] 87,123,32,-92,-121,-25,97,119,18,-101,-116,-11,106,113,4,-109,

[0850] 99,106,-35,-127,-43,100,105,-36,-127,-41,102,104,-38,-127,-40,103,

[0851] 110,76,-92,-97,69,114,-42,-124,13,127,17,-123,-46,112,72,-95,

[0852] 118,36,-124,-18,126,-1,-126,20,123,-38,-118,55,109,-72,-99,86,

[0853] 124,-9,-119,72,81,-115,-21,126,-45,-102,99,50,-125,15,117,-77,

[0854] 126,-54,-78,123,-31,-96,116,-6,-111,103,20,-120,87,44,-125,67,

[0855] 123,-93,-11,107,-117,35,75,-126,76,34,-117,107,-12,-93,124,-56,

[0856] 115,-119,61,28,-103,124,-81,-4,87,-125,98,-21,-67,121,-112,45,

[0857] 99,-124,113,-68,4,62,-109,124,-103,52,15,-77,117,-122,91,-34,

[0858] 73,-104,121,-122,107,-78,39,5,-49,86,-112,124,-118,98,-64,22,

[0859] 40,-60,78,-94,107,-116,122,-123,121,-114,104,-90,73,-54,33,-11,

[0860] ], [

[0862] 8,21,33,44,56,67,77,86,95,103,110,115,120,124,126,127,

[0863] 25,59,89,112,125,127,119,100,73,41,4,-33,-67,-95,-115,-126,

[0864] 41,92,123,124,98,48,-12,-70,-112,-127,-113,-73,-16,44,95,124,

[0865] 56,115,123,76,-4,-83,-125,-111,-48,37,105,127,92,16,-67,-120,

[0866] 71,127,91,-9,-103,-123,-55,53,123,105,12,-89,-127,-73,33,115,

[0867] 85,124,35,-90,-122,-27,96,119,20,-101,-117,-12,105,113,4,-109,

[0868] 97,108,-31,-127,-46,99,106,-34,-127,-43,101,104,-37,-127,-40,103,

[0869] 108,79,-90,-100,66,115,-40,-124,11,127,19,-123,-47,112,73,-95,

[0870] 117,40,-123,-22,126,2,-127,17,124,-36,-118,54,110,-71,-99,86,

[0871] 123,-4,-121,68,85,-113,-25,126,-42,-104,97,52,-125,14,117,-77,

[0872] 126, -49, -83, 122, -26, -99, 114, -2, -112, 102, 22, -121, 86, 45, -126, 67,

[0873] 125, -89, -17, 110, -115, 30, 79, -126, 73, 37, -118, 106, -10, -94, 123, -56,

[0874] 118, -116, 55, 34, -106, 123, -78, -8, 89, -125, 97, -19, -69, 121, -111, 45,

[0875] 103, -124, 110, -63, -1, 66, -111, 124, -101, 49, 18, -79, 117, -122, 91, -33,

[0876] 78, -107, 122, -121, 105, -75, 35, 9, -52, 88, -113, 123, -118, 97, -64, 22,

[0877] 43, -62, 80, -96, 108, -117, 122, -123, 120, -113, 103, -89, 72, -53, 33, -11, ]

[0879] 32p LGT:

[0880] A 32p LGT kernel can be any 32×32 matrix from the following set: [

[0882] 25,30,35,40,45,50,55,59,64,68,72,76,80,84,88,91,94,97,100,103,106,108,110,112,114,115,117,118,119,119,120,120

[0883] 68,81,92,102,110,116,120,122,121,119,115,109,101,91,80,67,53,39,23,8,-8,-24,-39,-54,-68,-80,-92,-101,-109,-115,-119,-122,

[0884] 96,111,120,124,121,112,98,80,57,32,5,-23,-49,-72,-93,-108,-119,-123,-122,-115,-102,-84,-62,-37,-11,17,43,67,89,105,117,123,

[0885] 112,124,123,111,87,55,18,-21,-58,-89,-112,-124,-123,-111,-88,-56,-19,20,58,89,112,124,123,111,88,56,19,-20,-57,-89,-111,-124,

[0886] 121,125,109,75,28,-23,-70,-106,-124,-122,-100,-62,-13,38,82,113,126,118,90,48,-2,-52,-93,-119,-126,-111,-79,-33,18,66,103,123,

[0887] 125,119,83,27,-36,-90,-122,-123,-94,-42,21,79,117,126,104,56,-6,-66,-110,-127,-112,-69,-9,53,102,125,118,81,24,-39,-92,-123,

[0888] 127,108,51,-23,-90,-125,-116,-67,5,76,120,122,82,13,-61,-113,-126,-95,-31,44,104,127,106,48,-27,-92,-125,-115,-64,9,79,121,

[0889] 127,93,16,-69,-121,-117,-59,27,100,127,94,18,-67,-121,-118,-61,25,99,127,96,20,-65,-120,-119,-62,23,98,127,97,22,-64,-120,

[0890] 126,76,-19,-103,-125,-73,24,106,125,69,-28,-108,-124,-66,32,110,123,62,-36,-112,-122,-59,40,114,120,55,-43,-116,-119,-51,47,117,

[0891] 124,57,-53,-123,-102,-5,96,126,62,-47,-122,-106,-11,91,127,68,-41,-120,-109,-18,87,127,73,-36,-118,-112,-24,83,127,78,-30,-115,

[0892] 122,37,-82,-127,-58,63,127,77,-43,-124,-94,21,117,107,2,-106,-118,-24,91,125,46,-74,-127,-66,55,126,84,-34,-121,-100,11,112,

[0893] 119,16,-105,-113,-1,112,106,-13,-119,-97,28,123,87,-42,-126,-76,56,128,63,-69,-127,-50,80,125,36,-91,-121,-22,101,116,7,-109,

[0894] 116,-5,-120,-86,55,127,41,-96,-114,10,122,82,-60,-127,-36,100,111,-16,-123,-78,65,126,31,-103,-109,21,124,73,-69,-126,-26,106,

[0895] 113,-26,-127,-47,100,104,-41,-128,-32,109,95,-55,-126,-17,116,84,-69,-123,-2,122,71,-81,-118,14,126,58,-92,-111,29,127,44,-102,

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[1016] ], [

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[1050] ], [

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[1084] ], [

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[1118] ], [

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[1137] 124,-42,-114,68,98,-92,-77,110,51,-122,-22,127,-7,-125,37,116,-64,-101,88,81,-107,-55,120,27,-126,2,126,-32,-118,60,104,-84,

[1138] 121,-64,-94,104,50,-125,3,123,-56,-100,98,58,-123,-5,125,-48,-105,93,65,-120,-14,126,-40,-109,86,73,-117,-23,127,-31,-114,80,

[1139] 117,-83,-66,124,-10,-118,82,67,-123,8,118,-81,-69,123,-6,-119,79,70,-122,5,120,-78,-72,122,-3,-120,76,73,-121,1,121,-75,

[1140] 112,-100,-32,126,-67,-72,125,-27,-103,109,17,-122,80,59,-127,42,94,-116,-1,117,-92,-45,127,-56,-82,122,-14,-110,102,30,-125,70,

[1141] 106,-113,3,110,-109,-4,113,-106,-11,116,-102,-18,119,-97,-25,121,-92,-32,123,-87,-39,125,-82,-45,126,-76,-52,127,-71,-58,127,-65,

[1142] 99,-122,39,78,-127,66,52,-125,90,24,-116,108,-6,-101,121,-35,-81,127,-63,-56,126,-87,-28,118,-106,2,104,-119,31,84,-126,59,

[1143] 91,-126,71,35,-116,115,-31,-74,127,-89,-12,105,-122,53,55,-123,104,-10,-90,127,-72,-34,115,-115,32,73,-127,90,11,-104,123,-54,

[1144] 83,-127,98,-13,-79,126,-101,18,75,-125,104,-24,-71,124,-107,29,66,-123,110,-34,-62,122,-112,38,57,-120,115,-43,-53,119,-117,48,

[1145] 74,-122,117,-59,-24,97,-127,101,-30,-55,115,-124,78,2,-81,125,-113,51,33,-103,127,-95,21,63,-118,121,-71,-11,88,-126,108,-42,

[1146] 64,-114,126,-97,36,37,-98,126,-113,63,8,-76,120,-124,87,-22,-51,106,-127,106,-50,-22,87,-124,119,-76,7,64,-114,126,-97,36,

[1147] 54,-102,126,-121,88,-35,-26,81,-117,127,-107,62,-3,-56,103,-126,120,-86,33,28,-83,118,-126,105,-60,1,58,-104,126,-119,85,-31,

[1148] 44,-86,115,-127,120,-96,57,-9,-39,82,-113,127,-122,99,-61,14,34,-78,110,-126,123,-102,66,-20,-29,74,-108,125,-124,105,-70,25,

[1149] 33,-67,95,-115,126,-126,115,-95,67,-33,-4,40,-73,100,-118,127,-125,112,-90,60,-26,-11,47,-79,104,-121,127,-123,108,-85,54,-18,

[1150] 22,-46,68,-87,104,-116,124,-127,126,-119,109,-94,76,-55,32,-7,-17,41,-64,84,-101,114,-123,127,-126,121,-111,97,-80,59,-36,12,

[1151] 11,-23,35,-47,58,-69,79,-88,97,-104,111,-116,121,-124,126,-127,127,-125,123,-119,114,-108,101,-93,85,-75,65,-54,42,-31,18,-6,

[1152] ], [

[1154] 6,12,18,24,30,36,42,48,54,59,64,70,75,80,84,89,93,97,101,105,108,111,114,116,119,121,123,124,125,126,127,127,

[1155] 18,36,54,70,84,97,108,116,123,126,127,125,121,114,105,93,80,64,48,30,12,-6,-24,-42,-59,-75,-89,-101,-111,-119,-124,-127,

[1156] 30,59,84,105,119,126,126,119,105,84,59,30,-0,-30,-59,-84,-105,-119,-126,-126,-119,-105,-84,-59,-30,-0,30,59,84,105,119,126,

[1157] 42,80,108,124,126,114,89,54,12,-30,-70,-101,-121,-127,-119,-97,-64,-24,18,59,93,116,127,123,105,75,36,-6,-48,-84,-111,-125,

[1158] 54,97,123,125,105,64,12,-42,-89,-119,-127,-111,-75,-24,30,80,114,127,116,84,36,-18,-70,-108,-126,-121,-93,-48,6,59,101,124,

[1159] 64,111,127,108,59,-6,-70,-114,-127,-105,-54,12,75,116,126,101,48,-18,-80,-119,-125,-97,-42,24,84,121,124,93,36,-30,-89,-123,

[1160] 75,121,121,75,-0,-75,-121,-121,-75,0,75,121,121,75,-0,-75,-121,-121,-75,0,75,121,121,75,-0,-75,-121,-121,-75,0,75,121,

[1161] 84,126,105,30,-59,-119,-119,-59,30,105,126,84,-0,-84,-126,-105,-30,59,119,119,59,-30,-105,-126,-84,0,84,126,105,30,-59,-119,

[1162] 93,127,80,-18,-105,-124,-64,36,114,119,48,-54,-121,-111,-30,70,125,101,12,-84,-127,-89,6,97,126,75,-24,-108,-123,-59,42,116,

[1163] 101,123,48,-64,-126,-89,18,111,116,30,-80,-127,-75,36,119,108,12,-93,-125,-59,54,124,97,-6,-105,-121,-42,70,127,84,-24,-114,

[1164] 108,114,12,-101,-119,-24,93,123,36,-84,-125,-48,75,127,59,-64,-127,-70,54,126,80,-42,-124,-89,30,121,97,-18,-116,-105,6,111,

[1165] 114,101,-24,-123,-84,48,127,64,-70,-126,-42,89,121,18,-105,-111,6,116,97,-30,-124,-80,54,127,59,-75,-125,-36,93,119,12,-108,

[1166] 119,84,-59,-126,-30,105,105,-30,-126,-59,84,119,-0,-119,-84,59,126,30,-105,-105,30,126,59,-84,-119,-0,119,84,-59,-126,-30,105,

[1167] 123,64,-89,-111,30,127,36,-108,-93,59,124,6,-121,-70,84,114,-24,-127,-42,105,97,-54,-125,-12,119,75,-80,-116,18,126,48,-101,

[1168] 125,42,-111,-80,84,108,-48,-124,6,126,36,-114,-75,89,105,-54,-123,12,127,30,-116,-70,93,101,-59,-121,18,127,24,-119,-64,97,

[1169] 127,18,-124,-36,119,54,-111,-70,101,84,-89,-97,75,108,-59,-116,42,123,-24,-126,6,127,12,-125,-30,121,48,-114,-64,105,80,-93,

[1170] 127,-6,-127,12,126,-18,-125,24,124,-30,-123,36,121,-42,-119,48,116,-54,-114,59,111,-64,-108,70,105,-75,-101,80,97,-84,-93,89,

[1171] 126,-30,-119,59,105,-84,-84,105,59,-119,-30,126,0,-126,30,119,-59,-105,84,84,-105,-59,119,30,-126,-0,126,-30,-119,59,105,-84,

[1172] 124,-54,-101,97,59,-123,-6,125,-48,-105,93,64,-121,-12,126,-42,-108,89,70,-119,-18,127,-36,-111,84,75,-116,-24,127,-30,-114,80,

[1173] 121,-75,-75,121,0,-121,75,75,-121,-0,121,-75,-75,121,0,-121,75,75,-121,-0,121,-75,-75,121,0,-121,75,75,-121,0,121,-75,

[1174] 116,-93,-42,127,-59,-80,123,-18,-108,105,24,-124,75,64,-126,36,97,-114,-6,119,-89,-48,127,-54,-84,121,-12,-111,101,30,-125,70,

[1175] 111,-108,-6,114,-105,-12,116,-101,-18,119,-97,-24,121,-93,-30,123,-89,-36,124,-84,-42,125,-80,-48,126,-75,-54,127,-70,-59,127,-64,

[1176] 105,-119,30,84,-126,59,59,-126,84,30,-119,105,0,-105,119,-30,-84,126,-59,-59,126,-84,-30,119,-105,-0,105,-119,30,84,-126,59,

[1177] 97,-125,64,42,-119,111,-24,-80,127,-84,-18,108,-121,48,59,-124,101,-6,-93,126,-70,-36,116,-114,30,75,-127,89,12,-105,123,-54,

[1178] 89,-127,93,-6,-84,127,-97,12,80,-126,101,-18,-75,125,-105,24,70,-124,108,-30,-64,123,-111,36,59,-121,114,-42,-54,119,-116,48,

[1179] 80,-124,114,-54,-30,101,-127,97,-24,-59,116,-123,75,6,-84,125,-111,48,36,-105,127,-93,18,64,-119,121,-70,-12,89,-126,108,-42,

[1180] 70,-116,125,-93,30,42,-101,127,-111,59,12,-80,121,-123,84,-18,-54,108,-127,105,-48,-24,89,-124,119,-75,6,64,-114,126,-97,36,

[1181] 59,-105,126,-119,84,-30,-30,84,-119,126,-105,59,0,-59,105,-126,119,-84,30,30,-84,119,-126,105,-59,-0,59,-105,126,-119,84,-30,

[1182] 48,-89,116,-127,119,-93,54,-6,-42,84,-114,127,-121,97,-59,12,36,-80,111,-126,123,-101,64,-18,-30,75,-108,125,-124,105,-70,24,

[1183] 36,-70,97,-116,126,-125,114,-93,64,-30,-6,42,-75,101,-119,127,-124,111,-89,59,-24,-12,48,-80,105,-121,127,-123,108,-84,54,-18,

[1184] 24,-48,70,-89,105,-116,124,-127,125,-119,108,-93,75,-54,30,-6,-18,42,-64,84,-101,114,-123,127,-126,121,-111,97,-80,59,-36,12,

[1185] 12,-24,36,-48,59,-70,80,-89,97,-105,111,-116,121,-124,126,-127,127,-125,123,-119,114,-108,101,-93,84,-75,64,-54,42,-30,18,-6,

[1186] ], [

[1188] 6,12,18,24,30,36,42,48,53,59,64,69,74,79,84,89,93,97,101,105,108,111,114,117,119,121,123,124,125,126,127,127,

[1189] 17,35,52,68,83,96,107,116,122,126,127,126,121,114,105,94,80,65,49,31,13,-6,-24,-42,-59,-74,-89,-101,-111,-119,-124,-127,

[1190] 28,57,82,103,118,126,127,120,106,86,61,32,2,-29,-58,-83,-104,-118,-126,-126,-119,-105,-85,-60,-31,-1,30,59,84,105,119,126,

[1191] 39,77,106,123,127,115,91,56,15,-28,-68,-100,-120,-127,-120,-98,-66,-26,17,58,92,116,127,123,105,75,37,-6,-48,-84,-111,-125,

[1192] 49,94,121,126,107,68,16,-39,-86,-118,-127,-113,-77,-27,28,78,113,127,117,86,38,-17,-68,-107,-126,-121,-94,-49,6,59,101,124,

[1193] 59,108,127,111,64,-1,-66,-112,-127,-107,-57,9,72,115,127,103,51,-16,-78,-118,-126,-98,-44,23,83,121,124,94,37,-30,-89,-123,

[1194] 69,119,123,80,6,-70,-119,-122,-79,-5,71,120,122,78,4,-72,-120,-122,-77,-3,73,120,122,76,2,-74,-121,-121,-75,-1,74,121,

[1195] 78,125,109,37,-53,-116,-121,-64,25,101,127,88,5,-81,-126,-107,-34,56,118,120,62,-28,-103,-126,-86,-2,83,126,105,31,-59,-119,

[1196] 87,127,86,-11,-100,-125,-70,30,111,121,53,-49,-119,-113,-35,66,125,103,16,-82,-127,-91,4,96,126,76,-23,-107,-123,-60,42,117,

[1197] 95,125,56,-57,-125,-94,11,107,119,37,-75,-127,-79,31,117,110,17,-90,-126,-62,50,123,99,-4,-103,-121,-44,69,127,85,-24,-114,

[1198] 102,118,21,-95,-122,-33,87,124,43,-79,-126,-54,70,127,64,-60,-127,-73,50,126,82,-39,-123,-91,28,120,98,-17,-116,-105,6,111,

[1199] 109,107,-15,-120,-91,40,126,71,-63,-127,-49,84,123,24,-101,-114,1,115,100,-26,-123,-82,51,127,61,-73,-126,-38,92,119,13,-108,

[1200] 114,92,-50,-127,-40,99,109,-22,-125,-66,78,121,7,-116,-89,54,127,35,-102,-107,26,126,62,-82,-120,-2,118,85,-58,-126,-31,104,

[1201] 119,74,-81,-116,21,126,45,-103,-99,52,126,14,-118,-75,79,116,-19,-126,-47,102,100,-50,-126,-15,118,77,-78,-117,17,126,48,-101,

[1202] 123,53,-105,-88,76,113,-39,-126,-3,125,44,-110,-81,84,108,-48,-124,7,126,35,-115,-73,91,103,-57,-122,16,127,26,-118,-65,97,

[1203] 125,30,-121,-47,115,63,-106,-77,95,91,-83,-102,68,112,-53,-119,36,124,-19,-127,2,127,16,-125,-33,120,50,-113,-66,104,80,-93,

[1204] 127,6,-127,1,127,-8,-127,15,126,-22,-125,29,123,-35,-121,42,119,-48,-116,55,113,-61,-110,67,106,-73,-102,78,98,-84,-93,89,

[1205] 127,-19,-122,49,110,-76,-91,99,67,-115,-38,125,8,-127,24,121,-54,-108,80,88,-102,-63,117,34,-126,-3,126,-29,-119,58,105,-84,

[1206] 126,-43,-107,90,68,-119,-16,126,-39,-109,87,71,-118,-19,127,-36,-111,85,74,-117,-23,127,-33,-113,82,77,-116,-26,127,-29,-114,79,

[1207] 124,-65,-83,117,10,-123,67,82,-118,-9,123,-68,-81,118,7,-123,70,79,-119,-5,122,-71,-78,120,3,-121,73,76,-120,-1,121,-74,

[1208] 121,-85,-52,127,-50,-87,120,-9,-112,100,32,-125,69,70,-125,30,101,-111,-11,120,-86,-51,127,-51,-86,120,-10,-112,100,31,-125,69,

[1209] 116,-102,-17,118,-99,-22,120,-95,-27,121,-92,-32,123,-88,-36,124,-85,-41,125,-81,-46,126,-77,-51,126,-73,-55,127,-69,-60,127,-64,

[1210] 110,-115,21,91,-125,51,66,-127,78,38,-121,100,7,-108,116,-25,-88,125,-55,-63,126,-81,-33,120,-103,-2,105,-118,29,85,-126,59,

[1211] 103,-123,56,51,-121,107,-16,-85,127,-79,-25,111,-119,42,64,-125,98,-2,-96,125,-66,-39,118,-113,28,76,-127,88,13,-105,122,-53,

[1212] 95,-127,87,2,-90,127,-92,5,85,-127,97,-12,-79,126,-101,20,73,-125,106,-27,-67,123,-110,34,61,-121,113,-41,-54,119,-116,48,

[1213] 86,-125,110,-47,-37,105,-126,93,-18,-64,118,-121,70,11,-88,126,-109,44,40,-106,126,-91,16,66,-119,120,-68,-13,89,-126,108,-42,

[1214] 76,-119,124,-88,24,48,-104,127,-108,54,17,-83,122,-121,81,-15,-57,109,-127,103,-45,-27,90,-124,118,-73,5,65,-114,126,-97,36,

[1215] 64,-108,126,-116,80,-25,-35,88,-120,125,-102,55,4,-62,106,-126,117,-82,28,33,-86,119,-126,103,-58,-1,60,-105,126,-118,84,-30,

[1216] 52,-92,118,-127,117,-90,50,-2,-46,87,-115,127,-120,95,-56,10,39,-81,112,-126,122,-100,63,-17,-32,75,-108,125,-124,104,-69,24,

[1217] 40,-73,99,-118,126,-124,112,-91,62,-28,-9,44,-76,102,-119,127,-123,110,-87,58,-23,-14,49,-80,105,-121,127,-122,107,-84,53,-18,

[1218] 27,-50,72,-90,106,-117,124,-127,125,-118,107,-92,73,-52,29,-5,-20,43,-65,85,-101,114,-123,127,-126,120,-111,97,-79,59,-36,12,

[1219] 14,-26,37,-49,60,-70,80,-89,98,-105,111,-117,121,-124,126,-127,126,-125,122,-118,113,-107,101,-93,84,-74,64,-53,42,-30,18,-6,

[1220] ], [

[1222] 5,11,17,24,30,36,42,47,53,59,64,69,74,79,84,89,93,97,101,105,108,111,114,117,119,121,123,124,126,126,127,127,

[1223] 15,34,51,68,82,96,107,116,122,126,127,126,122,115,106,94,81,66,49,32,13,-5,-24,-42,-59,-74,-89,-101,-111,-119,-124,-127,

[1224] 26,55,81,102,117,126,127,120,107,87,62,34,3,-28,-57,-82,-103,-118,-126,-127,-120,-106,-86,-60,-32,-1,30,59,84,105,119,126,

[1225] 36,74,105,123,127,117,93,58,17,-26,-66,-98,-120,-127,-120,-100,-67,-28,15,57,91,116,127,123,106,76,38,-5,-47,-84,-111,-125,

[1226] 46,92,120,127,109,71,19,-36,-84,-117,-127,-114,-79,-30,26,76,112,127,118,87,39,-15,-68,-107,-126,-122,-94,-49,5,59,101,124,

[1227] 55,106,127,113,67,3,-62,-110,-127,-109,-60,5,69,114,127,104,53,-14,-76,-117,-126,-100,-45,22,83,120,125,94,38,-30,-89,-123,

[1228] 64,117,124,84,11,-66,-118,-124,-82,-9,68,118,123,81,7,-69,-119,-123,-79,-5,71,120,122,78,3,-73,-120,-122,-76,-1,74,121,

[1229] 73,124,112,43,-48,-114,-123,-69,20,99,127,91,9,-78,-125,-109,-37,53,117,121,64,-26,-102,-127,-87,-3,83,126,106,32,-59,-119,

[1230] 82,127,91,-4,-96,-126,-75,24,108,123,58,-44,-118,-115,-39,63,124,105,19,-80,-127,-93,1,94,127,77,-22,-107,-123,-60,42,117,

[1231] 90,126,63,-50,-124,-99,4,104,121,42,-70,-127,-83,26,115,113,21,-88,-126,-65,48,123,101,-1,-102,-122,-45,68,127,85,-24,-114,

[1232] 97,121,30,-90,-124,-40,82,126,50,-74,-127,-59,65,127,68,-56,-127,-77,46,125,85,-36,-123,-92,26,120,99,-16,-116,-106,5,111,

[1233] 104,112,-6,-117,-97,32,125,78,-57,-127,-55,79,124,30,-98,-116,-4,113,103,-23,-122,-85,48,127,63,-71,-126,-39,92,120,13,-108,

[1234] 110,99,-41,-127,-48,93,114,-14,-123,-72,73,123,13,-114,-93,49,127,40,-99,-109,23,125,65,-80,-121,-4,118,87,-57,-126,-31,104,

[1235] 115,82,-73,-120,11,125,53,-98,-104,44,127,21,-116,-80,75,119,-14,-126,-51,99,102,-47,-126,-18,117,78,-77,-118,16,126,49,-101,

[1236] 119,63,-99,-95,68,117,-30,-127,-11,123,51,-106,-86,79,111,-42,-125,2,126,39,-113,-76,88,105,-54,-122,14,127,27,-118,-65,97,

[1237] 123,41,-117,-56,110,71,-100,-84,89,96,-76,-106,62,115,-47,-121,31,125,-14,-127,-3,127,19,-124,-36,119,52,-112,-67,104,80,-93,

[1238] 125,17,-126,-10,127,2,-127,6,127,-13,-126,21,125,-28,-123,36,121,-43,-118,50,115,-57,-112,64,108,-71,-104,77,99,-83,-94,88,

[1239] 127,-7,-125,38,115,-68,-98,92,75,-111,-46,123,15,-127,17,123,-48,-111,76,91,-99,-66,116,37,-125,-5,127,-27,-120,57,105,-84,

[1240] 127,-31,-113,81,77,-115,-26,127,-30,-114,81,78,-115,-27,127,-30,-114,80,78,-115,-27,127,-29,-114,80,79,-115,-28,127,-28,-114,79,

[1241] 126,-54,-92,112,21,-126,58,89,-114,-18,125,-61,-86,116,14,-124,65,83,-117,-10,123,-68,-80,119,6,-122,71,77,-120,-2,121,-74,

[1242] 124,-76,-63,126,-40,-94,116,0,-116,94,40,-126,62,76,-124,24,104,-108,-16,122,-82,-55,127,-48,-88,119,-8,-113,99,32,-125,69,

[1243] 120,-94,-28,122,-91,-32,123,-89,-35,124,-86,-39,124,-83,-43,125,-80,-46,126,-77,-50,126,-74,-54,127,-71,-57,127,-67,-61,127,-64,

[1244] 115,-109,10,98,-122,42,74,-127,71,46,-123,95,14,-111,114,-19,-92,124,-50,-67,127,-78,-37,121,-101,-5,107,-117,28,85,-126,59,

[1245] 109,-120,46,60,-124,101,-7,-92,126,-72,-33,114,-116,36,69,-126,94,4,-99,125,-63,-43,119,-111,25,78,-127,86,15,-105,122,-53,

[1246] 102,-126,79,12,-96,127,-86,-3,90,-127,92,-6,-84,126,-98,14,77,-125,103,-23,-70,124,-108,31,63,-122,112,-39,-55,119,-116,48,

[1247] 93,-126,105,-38,-45,109,-125,87,-11,-70,121,-119,65,17,-91,126,-106,40,44,-108,126,-89,13,68,-120,119,-67,-15,90,-126,107,-42,

[1248] 82,-121,121,-82,17,54,-108,127,-104,49,23,-87,123,-119,77,-10,-60,111,-126,101,-42,-30,92,-125,117,-72,3,66,-114,126,-96,36,

[1249] 71,-111,127,-113,75,-19,-41,92,-122,124,-99,51,8,-66,108,-126,116,-79,25,36,-88,120,-125,102,-56,-3,61,-105,126,-118,83,-30,

[1250] 58,-96,120,-126,115,-86,45,3,-50,90,-117,127,-118,92,-53,6,41,-83,113,-126,121,-98,61,-15,-33,76,-109,125,-123,104,-69,24,

[1251] 44,-76,102,-119,126,-123,110,-88,59,-24,-12,47,-79,104,-120,126,-123,109,-86,56,-21,-15,50,-81,105,-121,127,-122,107,-83,53,-18,

[1252] 30,-53,74,-92,107,-118,124,-127,124,-117,105,-90,71,-50,27,-3,-21,45,-66,86,-102,114,-123,126,-125,120,-110,96,-79,58,-36,12,

[1253] 15, -27, 39, -50, 61, -71, 81, -90, 98, -105, 112, -117, 121, -124, 126, -127, 126, -124, 122, -118, 113, -107, 100, -92, 83, -74, 64, -53, 42, -30, 18, -6,

[1254] ],

[1255] 64p LGT:

[1256] A 64p LGT core can be any 64×64 matrix from the following set: [

[1258] 25,28,30,33,35,37,40,42,45,47,49,52,54,56,59,61,63,65,67,69,71,74,76,77,79,81,83,85,87,89,90,92,94,95,97,98,100,101,102,104,105,106,107,108,109,110,111,112,113,114,115,115,116,117,117,118,118,119,119,119,119,120,120,120

[1259] 67,74,80,86,91,96,101,105,109,112,115,117,119,120,121,122,121,121,120,118,116,113,110,106,102,98,93,87,82,76,69,63,56,49,41,34,26,18,10,3,-5,-13,-21,-29,-36,-44,-51,-58,-65,-72,-78,-84,-89,-94,-99,-104,-107,-111,-114,-116,-118,-120,-121,-122,

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[1270] 127,96,33,-41,-102,-127,-110,-55,19,86,124,120,75,5,-67,-116,-126,-93,-28,46,105,128,107,50,-24,-90,-125,-118,-71,1,72,119,125,89,22,-51,-108,-128,-104,-45,29,94,126,116,66,-6,-76,-120,-124,-85,-17,56,111,127,101,40,-35,-97,-127,-113,-61,12,81,122,

[1271] 126,89,16,-63,-117,-125,-83,-8,70,120,123,77,0,-76,-122,-120,-70,7,82,124,117,64,-15,-88,-126,-114,-57,23,93,127,110,50,-31,-99,-128,-106,-42,38,103,128,101,35,-46,-108,-127,-96,-27,53,112,127,91,20,-60,-115,-125,-85,-12,67,119,123,79,4,-73,-121,

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[1275] 122,54,-50,-121,-112,-29,73,127,97,3,-93,-127,-78,23,108,123,56,-47,-120,-113,-32,71,126,99,6,-91,-128,-80,20,107,124,59,-45,-119,-114,-35,68,126,101,9,-89,-128,-83,17,105,124,61,-42,-118,-116,-37,66,125,102,12,-87,-128,-85,14,104,125,64,-40,-117,

[1276] 121,44,-65,-126,-95,6,103,124,54,-55,-124,-102,-5,96,126,64,-45,-121,-108,-16,88,128,73,-35,-117,-114,-27,80,128,82,-24,-112,-118,-38,71,127,90,-13,-107,-122,-48,61,126,98,-2,-100,-125,-58,52,123,104,9,-93,-127,-68,41,120,110,20,-85,-128,-77,31,116,

[1277] 120,35,-78,-128,-73,40,122,104,2,-102,-123,-44,70,128,82,-30,-118,-110,-13,95,125,54,-61,-127,-90,20,114,115,23,-88,-127,-63,52,125,97,-10,-108,-119,-33,80,128,72,-42,-122,-103,-1,102,122,43,-71,-128,-81,32,119,109,11,-96,-125,-53,62,127,89,-22,-114,

[1278] 118,25,-91,-125,-48,72,128,70,-50,-126,-89,27,119,105,-3,-108,-117,-22,93,125,45,-75,-128,-67,53,126,87,-30,-120,-103,6,110,116,18,-95,-124,-42,77,128,64,-56,-127,-84,33,121,101,-9,-111,-114,-15,97,123,39,-80,-127,-62,59,127,82,-36,-122,-99,12,113,

[1279] 117,15,-102,-119,-20,98,121,26,-94,-123,-32,90,124,37,-86,-125,-42,82,126,48,-77,-127,-53,73,128,58,-68,-128,-63,63,128,68,-58,-128,-73,53,127,78,-47,-126,-82,42,125,86,-37,-124,-91,31,122,94,-26,-121,-98,20,119,102,-14,-116,-105,9,114,108,-3,-111,

[1280] 115,5,-111,-109,8,117,101,-21,-121,-93,34,125,83,-47,-127,-73,59,128,62,-70,-127,-50,80,126,38,-90,-122,-25,99,118,12,-107,-112,1,113,106,-14,-119,-98,27,123,89,-40,-126,-79,52,128,68,-63,-128,-57,74,127,45,-85,-124,-32,94,121,20,-102,-116,-7,110,

[1281] 113,-5,-118,-95,37,126,71,-66,-127,-43,91,120,12,-110,-106,20,123,85,-51,-128,-58,78,125,28,-101,-114,3,117,97,-35,-126,-73,64,127,45,-89,-121,-14,109,107,-18,-122,-86,49,128,60,-76,-125,-31,99,115,-1,-116,-98,33,126,75,-62,-128,-47,88,122,16,-108,

[1282] 112,-15,-123,-79,63,127,34,-101,-111,17,124,78,-64,-127,-33,102,110,-18,-124,-77,65,127,31,-103,-110,19,124,76,-66,-126,-30,103,109,-20,-124,-75,67,126,29,-104,-108,21,125,74,-69,-126,-28,105,108,-23,-125,-73,70,126,26,-106,-107,24,125,72,-71,-126,-25,106,

[1283] 110,-25,-127,-60,86,118,-7,-123,-75,72,124,11,-116,-89,57,127,29,-108,-101,40,128,46,-97,-111,22,126,62,-84,-119,4,122,78,-70,-124,-14,115,91,-54,-127,-32,106,103,-37,-128,-49,95,113,-19,-126,-65,82,120,-1,-121,-80,67,125,17,-114,-93,51,128,34,-105,

[1284] 108,-35,-128,-39,105,100,-47,-127,-26,112,91,-59,-126,-13,118,82,-71,-123,-0,122,71,-81,-118,13,126,60,-91,-112,26,127,48,-100,-105,39,128,35,-108,-97,51,127,22,-114,-88,63,125,9,-120,-78,74,121,-4,-124,-67,85,116,-18,-126,-56,94,110,-31,-128,-43,103,

[1285] 106,-44,-127,-17,119,75,-82,-115,27,128,35,-111,-89,68,122,-9,-126,-52,101,101,-52,-126,-9,122,68,-89,-111,35,128,26,-115,-82,75,119,-18,-127,-44,106,95,-60,-124,0,124,60,-95,-106,44,127,18,-119,-75,82,115,-27,-128,-35,111,89,-68,-122,9,126,52,-101,

[1286] 104,-53,-124,5,126,44,-109,-87,76,116,-31,-128,-19,120,66,-95,-103,55,124,-7,-127,-42,110,85,-77,-115,33,128,17,-121,-64,96,101,-57,-123,9,127,40,-111,-84,79,114,-34,-128,-15,122,62,-98,-100,59,123,-11,-127,-38,112,82,-80,-113,36,127,13,-122,-60,99,

[1287] 102,-62,-120,27,128,10,-125,-47,111,79,-88,-105,58,121,-22,-128,-15,123,51,-109,-83,84,107,-53,-123,17,128,20,-122,-56,106,87,-81,-110,49,124,-12,-128,-25,121,60,-103,-90,77,112,-44,-125,7,127,30,-119,-64,100,94,-73,-115,39,126,-2,-127,-35,117,69,-97,

[1288] 100,-70,-113,48,123,-24,-127,-0,127,25,-122,-49,113,71,-99,-90,82,106,-61,-118,38,125,-14,-128,-11,126,35,-119,-58,108,79,-92,-97,73,111,-52,-121,28,127,-3,-128,-21,123,45,-115,-68,102,87,-85,-104,64,116,-42,-124,18,128,7,-126,-32,120,55,-110,-76,95,

[1289] 97,-78,-105,68,111,-57,-117,46,121,-34,-125,22,127,-10,-128,-2,128,15,-126,-27,124,39,-120,-50,115,61,-109,-72,102,82,-94,-91,86,99,-76,-106,66,113,-55,-118,44,122,-32,-125,20,127,-7,-128,-5,127,17,-126,-29,123,41,-119,-53,114,64,-108,-74,101,84,-93,

[1290] 95,-86,-95,86,95,-86,-95,86,94,-86,-94,86,94,-87,-94,87,94,-87,-94,87,94,-87,-94,87,93,-88,-93,88,93,-88,-93,88,93,-88,-93,88,92,-88,-92,89,92,-89,-92,89,92,-89,-92,89,92,-89,-91,90,91,-90,-91,90,91,-90,-91,90,91,-90,-91,90,

[1291] 93,-93,-83,101,73,-108,-63,115,51,-120,-39,124,27,-126,-14,128,2,-128,11,127,-24,-124,36,121,-48,-116,60,110,-71,-103,81,95,-90,-86,99,76,-107,-65,113,54,-119,-42,123,30,-126,-17,127,5,-128,8,127,-21,-125,33,122,-45,-117,57,112,-68,-105,79,97,-88,

[1292] 90,-99,-71,113,48,-123,-24,127,-1,-127,26,122,-50,-112,72,98,-92,-80,107,59,-119,-35,126,10,-128,15,125,-40,-117,63,105,-84,-88,101,68,-114,-46,123,21,-128,4,127,-29,-121,53,111,-75,-96,94,77,-109,-56,120,32,-126,-7,128,-18,-124,43,116,-66,-103,86,

[1293] 88,-105,-57,122,21,-128,17,123,-53,-107,85,82,-109,-50,124,14,-128,24,121,-60,-103,90,77,-113,-44,125,7,-127,31,118,-66,-99,95,71,-116,-37,127,-1,-126,38,115,-72,-94,100,65,-119,-30,127,-8,-125,45,112,-78,-89,104,58,-121,-22,128,-15,-123,52,108,-84,

[1294] 85,-110,-42,127,-7,-124,56,102,-96,-64,121,17,-128,33,115,-78,-84,111,41,-127,9,123,-57,-101,97,63,-121,-16,128,-34,-114,79,83,-112,-40,127,-10,-123,58,100,-98,-62,122,14,-127,36,114,-80,-82,112,38,-127,12,123,-60,-99,98,61,-122,-13,127,-37,-113,81,

[1295] 83,-115,-27,128,-35,-111,89,67,-122,-8,126,-54,-100,102,50,-126,12,121,-71,-86,113,31,-128,31,113,-86,-71,121,12,-127,49,102,-99,-54,126,-7,-122,67,90,-110,-36,128,-26,-115,82,75,-119,-17,127,-45,-105,96,58,-125,2,124,-63,-93,108,40,-128,22,117,-79,

[1296] 80,-119,-11,125,-62,-89,114,23,-127,51,98,-108,-35,128,-39,-105,101,47,-128,28,111,-93,-58,126,-16,-117,84,68,-123,4,121,-74,-78,120,9,-125,64,87,-115,-21,127,-53,-96,109,33,-128,42,104,-102,-44,128,-30,-110,94,55,-126,18,116,-86,-66,124,-6,-121,76,

[1297] 77,-122,5,119,-85,-61,126,-24,-110,98,44,-128,42,99,-109,-25,127,-60,-86,118,6,-122,76,71,-124,13,116,-91,-54,127,-32,-106,103,36,-128,50,94,-113,-18,125,-67,-80,121,-2,-120,83,64,-126,21,112,-96,-47,128,-39,-101,108,29,-127,57,88,-117,-10,123,-74,

[1298] 75,-125,21,109,-104,-29,126,-68,-74,125,-21,-108,104,28,-126,68,74,-125,22,108,-105,-28,126,-69,-73,125,-22,-108,105,27,-126,69,73,-125,23,107,-105,-27,126,-70,-72,125,-23,-107,106,26,-126,70,72,-125,24,107,-106,-26,126,-71,-72,125,-25,-107,106,25,-126,71,

[1299] 72,-126,36,95,-118,6,113,-102,-26,124,-81,-55,128,-55,-81,124,-25,-103,113,6,-118,95,37,-127,72,65,-127,44,90,-121,13,109,-107,-18,122,-87,-48,128,-62,-75,126,-33,-98,116,-2,-115,100,29,-125,78,58,-128,51,84,-123,21,105,-111,-10,119,-93,-40,127,-68,

[1300] 69,-128,51,79,-126,40,89,-123,28,97,-119,15,105,-114,3,111,-108,-9,117,-101,-21,121,-93,-34,125,-84,-45,127,-74,-57,128,-64,-67,128,-53,-78,126,-42,-87,124,-30,-96,120,-17,-104,115,-5,-110,109,7,-116,102,19,-121,94,32,-124,86,43,-127,76,55,-128,66,

[1301] 66,-128,65,61,-128,71,55,-127,76,49,-127,81,43,-125,86,37,-124,91,31,-122,95,24,-120,99,18,-118,103,12,-115,107,5,-112,110,-1,-109,114,-8,-106,116,-14,-102,119,-21,-98,121,-27,-93,123,-33,-89,125,-39,-84,126,-46,-79,127,-52,-74,128,-57,-69,128,-63,

[1302] 63,-128,78,40,-123,97,16,-114,111,-9,-101,122,-34,-83,127,-58,-63,128,-79,-40,123,-97,-15,114,-112,10,100,-122,35,83,-127,59,62,-128,80,39,-123,98,14,-113,112,-11,-100,122,-36,-82,127,-60,-61,127,-81,-38,123,-99,-13,113,-113,12,99,-122,37,81,-127,60,

[1303] 61,-127,90,19,-113,116,-25,-86,127,-66,-49,124,-99,-6,106,-121,38,76,-128,77,36,-120,107,-7,-98,124,-50,-65,127,-87,-24,115,-114,20,89,-127,62,53,-125,96,11,-109,119,-33,-80,128,-73,-41,122,-104,2,101,-123,46,69,-128,83,28,-117,111,-15,-93,126,-58,

[1304] 58,-125,101,-3,-97,126,-63,-46,121,-108,16,88,-128,74,33,-117,115,-29,-78,128,-84,-21,111,-120,41,67,-127,94,8,-104,124,-53,-56,124,-102,5,96,-126,65,44,-121,109,-18,-87,128,-75,-32,116,-115,30,77,-128,86,19,-110,120,-43,-66,127,-95,-6,103,-124,55,

[1305] 55,-122,110,-25,-76,127,-95,1,94,-127,77,24,-109,123,-55,-48,120,-113,32,70,-126,100,-7,-90,128,-82,-17,106,-124,62,42,-118,116,-39,-64,125,-104,14,84,-128,87,10,-101,126,-68,-35,115,-119,45,58,-123,108,-21,-79,128,-92,-3,97,-127,74,28,-111,121,-52,

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[1307] 48,-115,123,-66,-24,102,-127,87,-1,-85,127,-103,26,65,-122,116,-50,-42,112,-124,72,17,-98,128,-92,8,80,-126,107,-33,-59,120,-119,56,35,-109,126,-78,-10,93,-128,96,-15,-74,125,-111,39,52,-117,121,-63,-29,105,-127,83,3,-88,128,-101,22,69,-123,114,-46,

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[1309] 42,-106,128,-100,32,48,-109,127,-96,26,54,-112,127,-91,20,59,-115,126,-87,14,64,-118,125,-83,8,70,-120,123,-78,2,75,-122,121,-73,-4,80,-124,119,-68,-10,84,-125,117,-62,-16,89,-126,114,-57,-22,93,-127,111,-52,-28,97,-128,108,-46,-34,101,-128,105,-40,

[1310] 39,-100,128,-112,59,14,-83,123,-122,80,-11,-62,114,-127,98,-36,-39,100,-127,112,-59,-14,82,-123,122,-80,11,61,-113,127,-98,36,38,-100,127,-112,60,13,-82,123,-122,81,-12,-61,113,-127,99,-37,-38,99,-127,113,-60,-13,81,-123,122,-81,12,60,-113,127,-99,37,

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[1312] 33,-87,121,-126,102,-53,-8,67,-111,128,-115,75,-17,-44,96,-124,124,-94,42,20,-77,116,-128,109,-65,5,56,-103,127,-120,85,-30,-32,87,-121,126,-102,54,7,-66,110,-128,115,-76,18,44,-95,124,-124,95,-43,-19,76,-116,128,-110,66,-6,-55,103,-127,120,-86,31,

[1313] 30,-80,115,-128,116,-83,33,23,-75,112,-128,119,-87,39,17,-69,109,-127,121,-92,45,10,-64,105,-126,123,-96,51,4,-58,101,-125,125,-100,57,-2,-52,97,-123,126,-104,63,-9,-47,93,-122,127,-108,68,-15,-40,88,-119,128,-111,74,-22,-34,84,-117,128,-114,79,-28,

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[1316] 20,-56,87,-110,124,-128,120,-102,76,-42,5,32,-66,95,-116,127,-126,115,-94,65,-30,-7,44,-77,103,-121,128,-124,109,-86,54,-18,-19,55,-86,110,-124,128,-120,103,-76,43,-6,-31,66,-95,116,-127,127,-116,95,-66,31,6,-43,76,-103,121,-128,124,-110,86,-55,19,

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[1322] ], [

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[1388] ], [

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[1454] ], [

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[1520] ], [

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[1586] ], [

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[1652] ], [

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[1718] ], [

[1720] 4,7,10,13,16,19,22,25,28,31,34,37,40,43,46,49,52,55,57,60,63,65,68,71,73,76,78,81,83,85,87,90,92,94,96,98,100,102,104,105,107,109,110,112,113,115,116,117,118,119,120,121,122,123,124,125,125,126,126,126,127,127,127,127,

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[1784] ], [

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[1807] 116,104,-11,-115,-105,10,115,105,-10,-115,-105,9,115,106,-9,-114,-106,8,114,106,-8,-114,-106,7,114,107,-7,-114,-107,6,113,107,-6,-113,-108,5,113,108,-5,-113,-108,4,112,108,-4,-112,-109,3,112,109,-2,-111,-109,2,111,110,-1,-111,-110,1,111,110,-0,-110,

[1808] 118,97,-30,-124,-84,47,127,69,-63,-127,-54,78,125,37,-92,-121,-19,103,114,1,-113,-105,17,120,93,-35,-125,-80,52,127,65,-68,-127,-49,82,124,32,-95,-119,-14,106,111,-4,-115,-102,22,122,90,-39,-126,-76,56,127,61,-72,-127,-44,86,123,27,-98,-117,-9,109,

[1809] 120,88,-48,-127,-58,79,124,23,-104,-110,13,121,87,-49,-127,-57,80,123,22,-105,-109,14,121,86,-50,-127,-56,81,123,21,-105,-109,15,121,86,-50,-127,-55,82,123,20,-106,-108,16,122,85,-51,-127,-54,82,123,19,-107,-108,17,122,84,-52,-127,-53,83,122,18,-107,

[1810] 122,78,-64,-126,-29,105,106,-26,-126,-67,76,123,15,-112,-98,40,127,55,-87,-119,-1,118,88,-53,-127,-42,97,113,-13,-123,-78,65,126,28,-105,-106,27,126,66,-77,-123,-14,112,97,-40,-127,-54,87,119,0,-118,-88,53,127,41,-97,-113,14,123,77,-66,-126,-27,106,

[1811] 124,68,-80,-120,2,121,77,-72,-123,-8,118,84,-63,-125,-18,113,92,-54,-127,-28,108,99,-44,-127,-38,103,105,-35,-127,-48,96,110,-25,-126,-57,89,115,-14,-124,-66,81,119,-4,-122,-75,73,122,6,-118,-83,65,125,16,-114,-90,56,127,26,-109,-97,46,127,36,-104,

[1812] 125,57,-94,-109,33,127,38,-106,-97,52,126,18,-116,-82,71,122,-3,-123,-65,87,114,-24,-127,-47,101,103,-44,-127,-27,112,89,-62,-124,-6,121,73,-80,-118,14,126,56,-95,-108,35,127,36,-107,-96,54,126,16,-117,-81,72,121,-5,-124,-64,88,113,-25,-127,-45,102,

[1813] 126,46,-105,-94,62,123,-5,-125,-53,101,99,-55,-125,-3,123,59,-96,-104,48,126,10,-121,-66,91,108,-41,-127,-18,119,72,-85,-112,34,127,25,-116,-79,80,115,-26,-127,-32,112,84,-74,-118,19,127,40,-109,-90,67,121,-11,-126,-47,105,95,-61,-123,4,125,54,-100,

[1814] 127,34,-115,-76,87,108,-47,-125,2,126,44,-109,-84,78,113,-37,-127,-9,123,55,-103,-92,69,118,-26,-127,-20,120,64,-96,-100,60,122,-15,-127,-31,116,74,-89,-106,50,124,-4,-126,-42,111,82,-80,-112,39,126,7,-124,-52,105,91,-72,-117,29,127,18,-121,-62,98,

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[1816] 127,9,-126,-31,120,52,-111,-71,99,89,-84,-103,66,114,-46,-122,25,127,-3,-127,-19,124,41,-117,-61,106,79,-92,-95,76,109,-57,-119,37,125,-15,-127,-7,126,29,-121,-50,112,70,-100,-87,85,102,-68,-114,48,122,-27,-126,5,127,17,-124,-39,118,59,-107,-78,94,

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[1819] 126,-28,-123,42,118,-56,-111,69,103,-81,-93,92,82,-102,-70,111,57,-117,-43,122,29,-126,-14,127,-2,-127,17,125,-31,-121,46,116,-60,-109,73,100,-84,-90,95,79,-105,-67,112,53,-119,-39,123,25,-126,-10,128,-6,-127,21,124,-35,-120,50,114,-63,-107,76,98,-87,

[1820] 125,-39,-116,65,102,-87,-84,105,61,-118,-36,126,9,-127,19,123,-45,-114,70,98,-91,-79,108,55,-120,-30,127,2,-127,25,122,-51,-110,75,94,-96,-74,111,50,-122,-23,127,-4,-126,31,119,-57,-107,80,90,-100,-68,114,44,-124,-17,127,-11,-125,38,117,-63,-104,85,

[1821] 123,-51,-108,84,81,-110,-47,124,8,-127,32,117,-68,-96,98,65,-118,-28,127,-12,-124,50,108,-84,-82,109,48,-124,-9,127,-31,-117,68,96,-98,-66,118,29,-127,11,124,-50,-108,83,82,-109,-48,124,9,-127,30,118,-67,-97,97,66,-118,-30,127,-10,-124,49,109,-83,

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[1850] ], [

[1852] 3,6,9,12,15,19,22,25,28,31,34,37,40,43,46,48,51,54,57,60,62,65,68,70,73,75,78,80,83,85,87,90,92,94,96,98,100,102,104,105,107,109,110,112,113,115,116,117,119,120,121,122,123,123,124,125,125,126,126,127,127,127,127,127,

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[1866] 83,126,109,40,-48,-113,-124,-75,9,90,127,104,31,-57,-117,-122,-68,19,96,127,98,22,-65,-121,-119,-60,28,102,127,92,12,-73,-123,-115,-51,37,107,126,85,3,-80,-125,-110,-43,46,112,125,78,-6,-87,-127,-105,-34,54,116,123,70,-15,-94,-127,-100,-25,62,120,

[1867] 87,127,98,15,-75,-125,-107,-31,62,122,115,46,-48,-116,-121,-60,34,109,125,73,-19,-100,-127,-85,3,90,127,96,12,-78,-126,-105,-28,65,123,113,43,-51,-117,-120,-57,37,110,124,70,-22,-102,-127,-83,6,92,127,94,9,-80,-126,-104,-25,68,123,112,40,-54,-119,

[1868] 92,127,85,-9,-98,-127,-78,19,104,125,70,-28,-109,-123,-62,37,113,121,54,-46,-117,-117,-46,54,121,113,37,-62,-123,-109,-28,70,125,104,19,-78,-127,-98,-9,85,127,92,0,-92,-127,-85,9,98,127,78,-19,-104,-125,-70,28,109,123,62,-37,-113,-121,-54,46,117,

[1869] 96,126,70,-34,-115,-117,-40,65,125,100,6,-92,-127,-75,28,112,120,46,-60,-124,-104,-12,87,127,80,-22,-109,-122,-51,54,123,107,19,-83,-127,-85,15,105,123,57,-48,-121,-110,-25,78,127,90,-9,-102,-125,-62,43,119,113,31,-73,-127,-94,3,98,126,68,-37,-116,

[1870] 100,124,54,-57,-125,-98,3,102,123,51,-60,-125,-96,6,104,123,48,-62,-126,-94,9,105,122,46,-65,-126,-92,12,107,121,43,-68,-127,-90,15,109,120,40,-70,-127,-87,19,110,119,37,-73,-127,-85,22,112,117,34,-75,-127,-83,25,113,116,31,-78,-127,-80,28,115,

[1871] 104,121,37,-78,-127,-70,46,123,98,-9,-109,-117,-28,85,127,62,-54,-125,-92,19,113,113,19,-92,-125,-54,62,127,85,-28,-117,-109,-9,98,123,46,-70,-127,-78,37,121,104,0,-104,-121,-37,78,127,70,-46,-123,-98,9,109,117,28,-85,-127,-62,54,125,92,-19,-113,

[1872] 107,116,19,-96,-123,-37,83,126,54,-68,-127,-70,51,126,85,-34,-122,-98,15,115,109,3,-105,-117,-22,94,123,40,-80,-127,-57,65,127,73,-48,-125,-87,31,121,100,-12,-113,-110,-6,104,119,25,-92,-124,-43,78,127,60,-62,-127,-75,46,125,90,-28,-120,-102,9,112,

[1873] 110,110,-0,-110,-110,0,110,110,-0,-110,-110,0,110,110,-0,-110,-110,0,110,110,-0,-110,-110,0,110,110,-0,-110,-110,0,110,110,-0,-110,-110,0,110,110,-0,-110,-110,0,110,110,-0,-110,-110,0,110,110,-0,-110,-110,-0,110,110,0,-110,-110,-0,110,110,0,-110,

[1874] 113,104,-19,-121,-92,37,125,78,-54,-127,-62,70,127,46,-85,-123,-28,98,117,9,-109,-109,9,117,98,-28,-123,-85,46,127,70,-62,-127,-54,78,125,37,-92,-121,-19,104,113,-0,-113,-104,19,121,92,-37,-125,-78,54,127,62,-70,-127,-46,85,123,28,-98,-117,-9,109,

[1875] 116,96,-37,-126,-68,70,126,34,-98,-115,3,117,94,-40,-127,-65,73,125,31,-100,-113,6,119,92,-43,-127,-62,75,125,28,-102,-112,9,120,90,-46,-127,-60,78,124,25,-104,-110,12,121,87,-48,-127,-57,80,123,22,-105,-109,15,122,85,-51,-127,-54,83,123,19,-107,

[1876] 119,87,-54,-127,-40,98,112,-15,-123,-75,68,125,25,-107,-104,31,126,62,-80,-122,-9,115,94,-46,-127,-48,92,116,-6,-121,-83,60,127,34,-102,-109,22,125,70,-73,-124,-19,110,100,-37,-127,-57,85,120,3,-117,-90,51,127,43,-96,-113,12,123,78,-65,-126,-28,105,

[1877] 121,78,-70,-123,-9,117,85,-62,-125,-19,113,92,-54,-127,-28,109,98,-46,-127,-37,104,104,-37,-127,-46,98,109,-28,-127,-54,92,113,-19,-125,-62,85,117,-9,-123,-70,78,121,-0,-121,-78,70,123,9,-117,-85,62,125,19,-113,-92,54,127,28,-109,-98,46,127,37,-104,

[1878] 123,68,-85,-115,22,127,48,-100,-104,43,127,28,-112,-90,62,124,6,-121,-73,80,117,-15,-126,-54,96,107,-37,-127,-34,109,94,-57,-125,-12,119,78,-75,-120,9,125,60,-92,-110,31,127,40,-105,-98,51,126,19,-116,-83,70,122,-3,-123,-65,87,113,-25,-127,-46,102,

[1879] 124,57,-98,-102,51,125,6,-123,-62,94,105,-46,-126,-12,121,68,-90,-109,40,127,19,-119,-73,85,112,-34,-127,-25,116,78,-80,-115,28,127,31,-113,-83,75,117,-22,-127,-37,110,87,-70,-120,15,127,43,-107,-92,65,122,-9,-126,-48,104,96,-60,-123,3,125,54,-100,

[1880] 125,46,-109,-85,78,113,-37,-127,-9,123,54,-104,-92,70,117,-28,-127,-19,121,62,-98,-98,62,121,-19,-127,-28,117,70,-92,-104,54,123,-9,-127,-37,113,78,-85,-109,46,125,0,-125,-46,109,85,-78,-113,37,127,9,-123,-54,104,92,-70,-117,28,127,19,-121,-62,98,

[1881] 126,34,-117,-65,100,92,-75,-112,46,124,-12,-127,-22,122,54,-107,-83,85,105,-57,-121,25,127,9,-125,-43,113,73,-94,-98,68,116,-37,-126,3,127,31,-119,-62,102,90,-78,-110,48,123,-15,-127,-19,123,51,-109,-80,87,104,-60,-120,28,127,6,-125,-40,115,70,-96,

[1882] 127,22,-123,-43,116,62,-105,-80,92,96,-75,-109,57,119,-37,-125,15,127,6,-126,-28,122,48,-113,-68,102,85,-87,-100,70,112,-51,-121,31,126,-9,-127,-12,125,34,-120,-54,110,73,-98,-90,83,104,-65,-115,46,123,-25,-127,3,127,19,-124,-40,117,60,-107,-78,94,

[1883] 127,9,-127,-19,125,28,-123,-37,121,46,-117,-54,113,62,-109,-70,104,78,-98,-85,92,92,-85,-98,78,104,-70,-109,62,113,-54,-117,46,121,-37,-123,28,125,-19,-127,9,127,0,-127,-9,127,19,-125,-28,123,37,-121,-46,117,54,-113,-62,109,70,-104,-78,98,85,-92,

[1884] 127,-3,-127,6,127,-9,-127,12,127,-15,-126,19,126,-22,-125,25,125,-28,-124,31,123,-34,-123,37,122,-40,-121,43,120,-46,-119,48,117,-51,-116,54,115,-57,-113,60,112,-62,-110,65,109,-68,-107,70,105,-73,-104,75,102,-78,-100,80,98,-83,-96,85,94,-87,-92,90,

[1885] 127,-15,-125,31,122,-46,-116,60,109,-73,-100,85,90,-96,-78,105,65,-113,-51,120,37,-124,-22,127,6,-127,9,126,-25,-123,40,119,-54,-112,68,104,-80,-94,92,83,-102,-70,110,57,-117,-43,123,28,-126,-12,127,-3,-127,19,125,-34,-121,48,115,-62,-107,75,98,-87,

[1886] 127,-28,-121,54,109,-78,-92,98,70,-113,-46,123,19,-127,9,125,-37,-117,62,104,-85,-85,104,62,-117,-37,125,9,-127,19,123,-46,-113,70,98,-92,-78,109,54,-121,-28,127,-0,-127,28,121,-54,-109,78,92,-98,-70,113,46,-123,-19,127,-9,-125,37,117,-62,-104,85,

[1887] 126,-40,-113,75,90,-104,-57,122,19,-127,22,121,-60,-102,92,73,-115,-37,126,-3,-125,43,112,-78,-87,105,54,-123,-15,127,-25,-120,62,100,-94,-70,116,34,-127,6,125,-46,-110,80,85,-107,-51,123,12,-127,28,119,-65,-98,96,68,-117,-31,127,-9,-124,48,109,-83,

[1888] 125,-51,-104,94,65,-121,-15,127,-37,-112,83,78,-115,-31,127,-22,-119,70,90,-107,-46,126,-6,-123,57,100,-98,-60,123,9,-126,43,109,-87,-73,117,25,-127,28,116,-75,-85,110,40,-127,12,122,-62,-96,102,54,-124,-3,125,-48,-105,92,68,-120,-19,127,-34,-113,80,

[1889] 123,-62,-92,109,37,-127,28,113,-85,-70,121,9,-125,54,98,-104,-46,127,-19,-117,78,78,-117,-19,127,-46,-104,98,54,-125,9,121,-70,-85,113,28,-127,37,109,-92,-62,123,0,-123,62,92,-109,-37,127,-28,-113,85,70,-121,-9,125,-54,-98,104,46,-127,19,117,-78,

[1890] 122,-73,-78,120,6,-123,68,83,-117,-12,125,-62,-87,115,19,-126,57,92,-112,-25,127,-51,-96,109,31,-127,46,100,-105,-37,127,-40,-104,102,43,-127,34,107,-98,-48,127,-28,-110,94,54,-126,22,113,-90,-60,125,-15,-116,85,65,-124,9,119,-80,-70,123,-3,-121,75,

[1891] 120,-83,-62,126,-25,-109,100,40,-127,48,94,-113,-15,124,-70,-75,123,-9,-116,90,54,-127,34,104,-105,-31,127,-57,-87,117,6,-122,78,68,-125,19,112,-96,-46,127,-43,-98,110,22,-125,65,80,-121,3,119,-85,-60,126,-28,-107,102,37,-127,51,92,-115,-12,123,-73,

[1892] 117,-92,-46,127,-54,-85,121,-9,-113,98,37,-127,62,78,-123,19,109,-104,-28,125,-70,-70,125,-28,-104,109,19,-123,78,62,-127,37,98,-113,-9,121,-85,-54,127,-46,-92,117,-0,-117,92,46,-127,54,85,-121,9,113,-98,-37,127,-62,-78,123,-19,-109,104,28,-125,70,

[1893] 115,-100,-28,124,-80,-54,127,-57,-78,125,-31,-98,116,-3,-113,102,25,-123,83,51,-1...

Claims

1. A decoding method, comprising: Receives an entropy-encoded bitstream including compressed video data; One or more solution quantization blocks are generated through a process, the process including: The bitstream is parsed; Decoding the entropy-encoded bitstream; and Dequantize each of the one or more blocks; Determine whether at least one of the height and width of the one or more dequantized blocks is greater than or equal to a predetermined threshold; and In response to at least one of the height or width of the one or more dequantized blocks being greater than or equal to the predetermined threshold, the dequantized blocks are transformed and encoded using a tuned line graph transform (LGT) kernel to perform direct matrix multiplication on each of the horizontal and vertical dimensions of the one or more dequantized blocks, wherein the tuned LGT kernel is a discrete cosine transform (DCT) or discrete sine transform (DST) derived from a generalized graph Laplace (GGL) matrix.

2. The method according to claim 1, wherein, The predetermined threshold has a value corresponding to one of 32 and 64.

3. The method according to claim 1, wherein, The predetermined threshold has a value corresponding to one of 4, 8, 16, 32, 64, 128, and 256.

4. The method according to claim 1, further comprising: Before using the LGT core, the LGT core is generated using at least one of a first self-loop weight or a second self-loop weight, each of which is pre-set to a specific corresponding value.

5. The method according to claim 4, wherein, The discrete cosine transform or discrete sine transform is derived from DCT-2, DCT-8 or DST-7 derived from a generalized graph Laplace (GGL) matrix of a certain form.

6. The method according to claim 5, wherein, The method further includes at least one of the following: DCT-2 is derived by setting the first self-loop weight to 0; DST-7 is derived by setting the first self-loop weight to a first predetermined value; DCT-8 is derived by setting the second self-loop weight to the first predetermined value; DST-4 is derived by setting the first self-loop weight to twice the first predetermined value; and DCT-4 is derived by setting the second self-loop weight to twice the first predetermined value.

7. The method according to claim 4, wherein, Generating the LGT core includes: LGT kernels of different sizes are tuned using the first self-loop weights and the second self-loop weights to derive the first transform kernel; and A second transformation kernel is derived by brute force tuning one or more individual weights of one or more basis vectors of the first transformation kernel, wherein the second transformation kernel is a tuned LGT kernel.

8. The method of claim 1, further comprising: Determine whether the line graph transformation (Np LGT) of size N points is enabled; as well as When the Np LGT is enabled, only Discrete Cosine Transform (DCT)-2 and LGT are used as candidates for the transformation type of the edge of length N of the solution block.

9. The method of claim 1, further comprising: Determine whether the line graph transformation (Np LGT) of size N points is enabled; as well as When the Np LGT is enabled, only Discrete Cosine Transform (DCT)-2, LGT, and Flipped LGT are used as candidates for the transformation type of the edge of length N of the solution block.

10. The method of claim 1, further comprising: Determine whether the line graph transformation (Np LGT) of size N points is enabled; as well as When the Np LGT is enabled, identity transformations are avoided on the edges of the dequantized block with length N.

11. The method of claim 1, further comprising: Determine whether the line graph transformation (Np LGT) of size N points is enabled; as well as When the Np LGT is enabled, supported transform types are used for intra-frame prediction residuals and inter-frame prediction residuals that are different from each other.

12. The method according to claim 11, wherein, For the intra-frame prediction residual, the supported transform types include Discrete Cosine Transform (DCT)-2 and LGT.

13. The method according to claim 11, wherein, For the inter-frame prediction residual, the supported transform types include Discrete Cosine Transform (DCT)-2, LGT, and Flipped LGT.

14. The method of claim 1, further comprising: A displayable image is generated, at least in part, based on the transformed dequantized block; as well as Execute controls to display the displayable image on the monitor.

15. A non-transitory computer-readable storage medium storing instructions that, when executed, cause a system or device comprising one or more processors to: Receives an entropy-encoded bitstream including compressed video data; One or more solution quantization blocks are generated through a process, the process including: The bitstream is parsed; Decoding the entropy-encoded bitstream; and Dequantize each of the one or more blocks; Determine whether at least one of the height and width of the one or more dequantized blocks is greater than or equal to a predetermined threshold; and In response to at least one of the height or width of the one or more dequantized blocks being greater than or equal to the predetermined threshold, the dequantized blocks are transformed and encoded using a tuned line graph transform (LGT) kernel to perform direct matrix multiplication on each of the horizontal and vertical dimensions of the one or more dequantized blocks, wherein the tuned LGT kernel is a discrete cosine transform (DCT) or discrete sine transform (DST) derived from a generalized graph Laplace (GGL) matrix.

16. The non-transitory computer-readable storage medium according to claim 15, wherein, The instructions are further configured to cause the system or device including one or more processors to: At least in part, a displayable image is generated based on the transformed dequantized block; and Execute controls to display the displayable image on the monitor.

17. A decoding device, comprising: At least one memory configured to store computer program code; as well as At least one processor configured to access the at least one memory and operate according to the computer program code, the computer program code comprising: The receiving code is configured to cause the at least one processor to receive an entropy-coded bitstream comprising compressed video data; Generate code configured to cause the at least one processor to generate one or more dequantized blocks through a process comprising: parsing the bitstream, decoding the entropy-coded bitstream, and dequantizing each of the one or more blocks; Determine the code, configured to cause the at least one processor to determine whether at least one of the height and width of the one or more dequantized blocks is greater than or equal to a predetermined threshold; and Transform encoding code, configured such that the at least one processor: in response to at least one of the height and width of the one or more dequantized blocks being greater than or equal to the predetermined threshold, performs transform encoding on the dequantized blocks using a tuned line graph transform (LGT) kernel to perform direct matrix multiplication on each of the horizontal and vertical dimensions of the one or more dequantized blocks, wherein the tuned LGT kernel is a discrete cosine transform (DCT) or discrete sine transform (DST) derived from a generalized graph Laplacian (GGL) matrix.

18. The apparatus according to claim 17, wherein, The predetermined threshold has a value corresponding to one of 32 and 64.

19. The apparatus according to claim 17, wherein, The transformation encoding code is further configured to cause the at least one processor to: The tuned LGT core is generated in the following way: The LGT kernels of different sizes are tuned to specific values ​​using the first self-loop weight and the second self-loop weight to derive the first transform kernel; as well as A second transformation kernel is derived by brute force tuning one or more individual weights of one or more basis vectors of the first transformation kernel, wherein the second transformation kernel is a tuned LGT kernel.

20. The apparatus of claim 17, further comprising: A display, wherein the computer program code further includes display code, the display code being configured to cause the at least one processor to: generate a displayable image based at least in part on a transformed dequantized block; and perform control to display the displayable image via the display.

Images