Method for transmitting a message from a transmitting unit to a receiving unit

Abstract

Method (100) for transmitting a message from a transmitting unit (10) to a receiving unit (20) via a communication channel, the method (100) comprising the following steps: - determining (110) a basis matrix representing a protograph; - determining (120) an LDPC code using the previously determined basis matrix and the MacKay-Neal code generation method; - coding (130) the message to be transmitted using the previously determined LDPC code; and - transmission (140) of the coded message from the transmitting unit (10) to the receiving unit (20); - wherein determining (110) the base matrix comprises: - defining (112) a value range for a target code rate, the value range being defined by a lower range value R1 and an upper range value R2; - Determine (114) of n BKBasis matrix candidates, where n BK ≥ 2; - Calculate (116) the quality for each of the n BK Basis matrix candidates for at least two code rates each within the previously defined value range; - Selecting (118) the basis matrix candidate with the highest quality from the n BK Base matrix candidates; and - Set the base matrix to the following value: B 1 / 2 = [ 3 3 0 1 1 0 0 1 | 3 0 0 0 2 1 0 0 3 0 1 0 2 0 0 1 ] .

Description

[0001] The present invention relates to a method for transmitting a message from a transmitting unit to a receiving unit via a communication channel, as well as to a corresponding communication system. The present invention can be used in particular in a satellite communication system.

[0002] In Zahr et al. “Rate-Adaptive Protograph MayKay-Neal Codes”, 2023 IEEE Information Theory Workshop (ITW), Saint-Malo, France, 2023, pages 412-417, doi: 10.1109 / ITW55543.2023.10161636, URL: https: / / ieeexplore.ieee.org / stamp / stamp.jsp?tp=&arnumber=10161636 (downloaded on May 7, 2024) describes a method for transmitting a message from a transmitting unit to a receiving unit. This method proposes and analyzes the use of a rate-adaptive MacKay-Neal code.

[0003] The present invention specifically encompasses the development and use of novel LDPC codes (also referred to as low-density parity check codes) developed for a rate-adaptive MacKay-Neal scheme with binary phase shift keying (also referred to as binary phase shift keying or BPSK). The development and use of these novel codes makes it possible to operate with a near-biAWGN channel capacity (also referred to as binary input additive white Gaussian noise channel capacity) over a wide range of code rates.

[0004] LDPC codes (see [1] and [2]) are known from the state of the art and refer to error-correcting codes that are widely used in modern communication systems. They are used in particular to improve the reliability and efficiency of data transmission over noisy channels. LDPC codes are usually designed for a specific code rate (i.e., for a specific channel condition), which complicates the use of these codes in many practical communication systems where the signal-to-noise ratio (SNR) varies during data transmission. This is even more true for satellite communications in low Earth orbit (LEO), where channel characteristics change rapidly due to the relative motion between the satellites and a ground station.

[0005] To address this challenge, rate-adaptive schemes for LDPC codes have been developed in the past. Modern communication standards that aim for rate flexibility often specify a large number of LDPC control matrices (also called LDPC parity check matrices) to serve different data rates (see, e.g., [3]). Other common techniques for developing rate-compatible schemes are known as shortening, lengthening, and puncturing (see [4] and [5]). 5G-like prototype LDPC code designs (see [6] to [8]) typically rely on these techniques to change the code rate depending on channel quality.

[0006] However, the methods known from the state of the art have some disadvantages.

[0007] Implementing a variety of channel codes to cope with large variability in the signal-to-noise ratio (SNR) presents a particular challenge, especially in the case of high-throughput communication (e.g., free-space optical downlinks). To support such high data rates, the entire communication scheme must be as simple as possible. In rate-adaptive schemes based on shortening, lengthening, and puncturing techniques, the code rate is adjusted based on the current channel quality. This means that the amount of transmitted information varies over time, resulting in a non-constant block length. This can be problematic for frame synchronization. The solution presented in [5] achieves rate adaptation without changing the block length by using additional degree-2 nodes.However, the code developed for this scheme may have some limitations in terms of rate flexibility compared to other rate-adapting schemes based on shortening, lengthening, and puncturing schemes.

[0008] Based on the prior art described above, it is an object of the present invention to provide a method for transmitting messages that is particularly efficient and suitable for use in applications with varying channel quality.

[0009] To achieve the stated object, the present invention proposes a method for transmitting a message from a transmitting unit to a receiving unit via a communication channel according to independent claim 1. The method according to the invention comprises the following steps: - Determine a basis matrix that represents a protograph; - Determining an LDPC code using the previously determined basis matrix and the MacKay-Neal code generation method; - encoding the message to be transmitted using the previously determined LDPC code; and - Transmission of the coded message from the transmitting unit to the receiving unit; - where determining the base matrix comprises: - Specifying a range of values ​​for a target code rate, the range of values ​​being defined by a lower range value R1 and an upper range value R2; - Determine n BK Basis matrix candidates, where n BK ≥ 2; - Calculate the quality for each of the n BK Basis matrix candidates for at least two code rates each within the previously defined value range; - Selecting the basis matrix candidate with the highest quality from the n BK Base matrix candidates; and - Set the base matrix to the following value: B1 / 2=[33011001|3000210030102001].

[0010] The method according to the invention enables particularly efficient communication between a transmitting unit and a receiving unit via a communication channel having time-varying channel properties. Unlike the methods known from the prior art, the method according to the invention can use a single LDPC code to ensure reliable communication between the transmitting unit and the receiving unit, even with changing channel properties. For this purpose, a value range for the target code rate is defined in advance, which specifies the code rates at which the coding is to be performed. The value range of the target code rate specifies the code rates at which the message is to be encoded. This value range describes the expected channel quality or channel property.If particularly good channel properties are expected in a specific application, this value range can be selected close to 1, for example, [0.8, 0.9]. If relatively poor channel properties are expected in another application, the value range can be selected close to 0.5 (for example, [0.5, 0.6]) or even lower (for example, [0.3, 0.4]). If a strong variation in channel quality is expected, a relatively broad value range can be selected (for example, [0.4, 0.9]), whereas the specified value range can be selected relatively narrowly if no strong variation in channel quality is expected in a specific application (for example, [0.8, 0.85]).

[0011] A protograph (also called a protogram or protograph in English) is a small graph model used to generate LDPC codes. A protograph can be represented by a basis matrix, which in turn is a relatively small matrix compared to a control matrix (also called a parity check matrix).

[0012] The MacKay-Neal code generation method is a prior art method for generating LDPC codes using a basis matrix. It can be shown that the quality of the generated LDPC code depends significantly on the quality of the basis matrix. The method according to the invention ensures optimization of the basis matrix, thereby enabling improved quality of the generated LDPC codes.

[0013] As explained above, in the method according to the invention, a value range for the desired code rate (target code rate) is first defined, whereby this value range is defined by the values ​​R1 and R2. This value range is determined depending on the expected channel quality. Subsequently, n BK Base matrix candidates are determined, where n BKis at least 2, but preferably greater than 10 or 100. Various approaches can be chosen for determining the basis matrix candidates. For example, it can be provided that the basis matrix candidates are generated using a random generator. Alternatively, it can be provided that the basis matrix candidates are read from a previously provided database. The basis matrix candidates represent possible candidates that are suitable for the basis matrix. The basis matrix candidates are then each checked to see how well they are suited for transmission at given code rates. For this purpose, the suitability of the basis matrix candidates is checked for at least two code rates within the specified value range. For example, the quality can be calculated for the lower range value R1 and the upper range value R2 of each of the basis matrix candidates.It may also be provided that the quality of the respective base matrix is ​​calculated for three or more than three code rates within the specified value range. The quality is calculated for each of the n code rates. BKpossible basis matrix candidates are calculated. Various approaches can be chosen for calculating the quality of the basis matrix candidates. In particular, the density evolution (DE) analysis or the PEXIT analysis (protograph EXIT or protographic extrinsic information transfer analysis) can be used. For example, the PEXIT analysis calculates which data rate can be achieved for a given communication channel with a certain basis matrix for a given code rate. The data rate achievable with a basis matrix for a specific code rate can, for example, be compared with the theoretically achievable data rate (also known as the Shannon limit). The smaller the difference between the data rate achieved with a basis matrix and the theoretically achievable data rate, the higher the quality of a basis matrix at a given code rate.Once the quality for a basis matrix candidate has been calculated for a first code rate within the specified value range, the calculation for a second code rate within the value range can be performed. Subsequently, for example, an average value can be calculated from the difference between the data rate achieved with the first code rate and the theoretically achievable data rate, as well as between the data rate achieved with the second code rate and the theoretically achievable data rate, and the basis matrix candidate with the highest quality can be selected from the n. BK Basis matrix candidates can be selected, for example, by selecting the basis matrix for which the average difference between the achieved data rate and the theoretically achievable data rate for the two code rates (i.e. d(R1+d)+d(R2))2 ) is the smallest. This mean value can also be determined for three or more than three code rates. Alternatively, the basis matrix can be selected for which the product of the two (or more than two) distances is the smallest. It can also be provided that the selection of the basis matrix candidate with the highest quality is carried out in such a way that the basis matrix is ​​selected for which the maximum distance between the achieved data rate and the theoretical limit calculated for all code rates examined is minimal. In other words, the basis matrix is ​​selected for which the maximum deviation of the achieved data rate from the Shannon limit is the smallest.

[0014] While the calculation of the quality was described above, in particular based on two code rates within the value range, according to some embodiments of the present invention, it can be provided that the quality is calculated for three or more than three code rates within the specified value range. Analogous to the embodiment described above, it can be provided that the base matrix is ​​selected for which the highest quality is achieved for three or more than three code rates.

[0015] The method according to the invention can preferably be used in a satellite communications system, with the method preferably being used in a LEO satellite communications system. In satellite communications systems, a particularly significant improvement in transmission efficiency is achieved, since such systems have a particularly fluctuating communication channel quality. Therefore, the method according to the invention is particularly well suited for satellite communications, and in particular for LEO systems.

[0016] It can also preferably be provided that the calculation of the quality of the n BK Basis matrix candidates for at least two code rates have the following step: - Conduct a PEXIT analysis for each of the n BK Base matrix candidates for evaluating the quality of each of the n BK Base matrix candidates.

[0017] PEXIT analysis (protograph extrinsic information transfer analysis) is a state-of-the-art analysis technique that enables the performance analysis of protograph-based LDPC codes. The PEXIT technique is an extension of the EXIT technique used to analyze the performance of iterative decoders for LDPC codes. The PEXIT technique is based on the concept of protographs. The EXIT function of a protograph is a measure of how much information can be transferred between the decoder and the channel during each iteration when the protograph is used as a building block of an LDPC code. The PEXIT diagram is created by recording the EXIT functions of the protographs that comprise an LDPC code. The PEXIT diagram can then be used to determine the number of iterations the decoder requires to achieve a certain level of performance.The PEXIT graph can also be used to compare the performance of different prototypical LDPC codes. The PEXIT analysis method is a useful tool for analyzing the performance of prototypical LDPC codes. It is relatively simple to implement and can be used to predict the performance of an LDPC code across a variety of channels.

[0018] However, as already stated above, the present invention is not limited to a specific analysis method, but can also be used with other analysis methods as long as the analysis methods allow to evaluate the quality of a protograph or a basis matrix and to provide a numerical value that describes the quality for a basis matrix.

[0019] Furthermore, it can be provided that the calculation of the quality of the n BK Basis matrix candidates for at least two code rates have the following steps: - Calculate the transmission quality associated with each of the n BK Basis matrix candidates are achievable for at least two code rates, and - Determine the distances between the respective transmission quality associated with each of the n BK Basis matrix candidates for each of the at least two code rates achievable, with the transmission quality theoretically achievable for the code rates.

[0020] If, for example, ten basis matrix candidates have been determined and a total of three code rates are to be considered, a total of 30 calculations are carried out for the specified distance. Three distance values ​​are calculated for each of the ten basis matrix candidates, with a numerical value for the candidate's rating being determined from the three values ​​for each candidate. As explained above, this numerical value can be calculated, for example, by averaging the three individual values ​​(arithmetic mean or geometric mean) or by determining the maximum value. The total of ten numerical values, which represent the quality for each candidate, are then compared with one another, and the candidate with the highest quality is selected. For example, the candidate with the lowest numerical value can be selected.

[0021] According to some embodiments of the method according to the invention, it can be provided that the selection of the basis matrix candidate with the highest quality from the n BK basis matrix candidates in such a way that the basis matrix candidate is selected which has the smallest mean distance between the transmission quality associated with each of the n BKBasis matrix candidates are achievable for at least two code rates, and the theoretically achievable transmission quality has been determined. The mean distance can be determined by calculating an arithmetic or geometric mean of the individual values, whereby two, three or more than three individual values ​​can be calculated beforehand for each basis matrix candidate, depending on how many code rates within the specified value range are to be taken into account. In general, the more code rates are taken into account, the better the determined basis matrix. However, the more code rates are taken into account, the more calculation complexity increases. In this respect, when selecting the number of code rates to be taken into account, there is always a compromise between the quality to be achieved from the determined basis matrix and the required computational effort.Initial investigations have shown that considering two, three or four data rates within the value range represents a good compromise between a sufficiently high quality of the basis matrix and an acceptable computational effort for determining the basis matrix.

[0022] In the method according to the invention, it can also be provided that the calculation of the quality of the n BK Basis matrix candidates for at least two code rates within the previously defined range, calculating the quality of the n BK Basis matrix candidates for at least three code rates within the specified range, wherein the first code rate is equal to R1, the second code rate is equal to R2 and the third code rate is greater than R1 and less than R2, and preferably (R1+R2) / 2.

[0023] In the investigations underlying the present invention, it was shown that this approach can be used to determine basis matrices that lead to particularly good LDPC codes when the MacKay-Neal code generation method is used.

[0024] Furthermore, the method according to the invention can provide that the determination of n BK Basis matrix candidates involve the use of a random generator designed to generate random basis matrices. This provides a particularly efficient transmission method.

[0025] In addition, the method according to the invention can provide that the determination of n BKThe method for determining the basis matrix candidates comprises limiting the values ​​of the basis matrix candidates to a predetermined threshold, where the threshold is preferably 3. This allows the search area to be restricted, so that fewer basis matrix candidates need to be examined overall. This significantly reduces the computational effort without noticeably compromising the quality of the determined basis matrix. Overall, the efficiency of the method according to the invention is increased because the determination of the LDPC codes is significantly simplified.

[0026] Furthermore, the method according to the invention can provide that the n BK Base matrix candidates have m rows and n columns, with the first three values ​​of the first row of the base matrix candidates being set to the value 3. This restricts the base matrix candidates to a predefined structure, which has proven particularly useful in initial studies.

[0027] Furthermore, the method according to the invention can provide that the determination of the LDPC code using the previously determined basis matrix and the MacKay-Neal code generation method comprises a concatenation of an inner code and an outer code.

[0028] According to preferred embodiments of the method according to the invention, it can also be provided that the method additionally comprises the following steps: - Measurement of channel quality over a specified measurement period, - Specifying the value range for the desired code rate depending on the channel quality determined during the measurement period.

[0029] While in the embodiments of the invention described above the range was defined "a priori" and a fixed LDPC code was then determined and used, in this exemplary embodiment the channel quality can be monitored "online" or in real time and the range for the desired code rate can be adjusted depending on changes in the channel quality and a correspondingly adjusted LDPC code can be determined. This makes it possible to react in real time to changing properties of a communication channel and to provide the optimal LDPC code even with strongly varying channel quality. In this case the range is defined directly in response to the measured channel quality. As explained above, in the case of particularly good channel quality a high code rate orA range of values ​​for the target code rate is set close to 1, while for low channel quality, a range of values ​​for the target code rate is set close to 0.5 or lower. This can be achieved, for example, by using a lookup table (LUT) that stores different code rates depending on different channel quality values.

[0030] Furthermore, to achieve the object described above, a communication system is proposed, wherein the communication system comprises a transmitting unit, a receiving unit and a computing unit, and wherein the computing unit is designed to - to determine a basis matrix that represents a protograph; - to determine an LDPC code using the previously determined basis matrix and the MacKay-Neal code generation method; - where determining the base matrix comprises: - Specifying a range of values ​​for a target code rate, the range of values ​​being defined by a lower range value R1 and an upper range value R2; - Determine n BK Basis matrix candidates, where n BK ≥ 2; - Calculate the quality of the n BK Basis matrix candidates for at least two code rates within the previously defined range of values; - Selecting the basis matrix candidate with the highest quality from the n BK Base matrix candidates, and - Set the base matrix to the following value: B1 / 2=[33011001|3000210030102001]. and wherein the transmitting unit is designed to - encode the message using the previously determined LDPC code; and - to transmit the coded message to the receiving unit.

[0031] The computing unit of the communication system according to the invention can be designed to carry out all of the method steps described above in connection with the method according to the invention. According to one embodiment, the computing unit can be configured as part of the transmitting unit. Alternatively, the computing unit can be configured as a separate unit. Furthermore, the computing unit can be configured as a remote server, and the transmitting unit can access the remote server and be configured to receive the determined base matrix or the determined LDPC code from the remote server. Further preferred embodiments and features of the invention

[0032] One of the objectives of the present invention is to provide a message transmission method for a rate-adaptive scenario using a single LDPC code based on the MacKay-Neal (MN) construction. MN codes were introduced in [9] (Section VI) as a class of error-correcting codes based on sparse matrices for non-uniform sources. They can be considered as multi-edge LDPC codes ([1],[2]) defined by an extended bipartite graph with a set of variable nodes (VNs) associated with the source bits, while the remaining VNs are associated with the codeword bits. LDPC code constructions closely related to MN codes were proposed in

[10] and

[11] for joint source-channel coding.Originally, MN was introduced for non-uniform sources, but it was pointed out in [9] that MN can also be used with uniform sources by introducing a nonlinear block code that transforms the uniform output sequence of the source into a sequence with a prescribed distribution. [9] also recognized that this option can be interesting from the point of view of rate flexibility, since the rate of the scheme can be modified (e.g., to adapt to varying channel conditions) by changing the statistics of the sequences generated by the nonlinear block coder without changing the underlying LDPC code. This approach is interesting from two points of view. First, the possibility of rate adaptation without changing the underlying LDPC encoder / decoder is important to limit implementation complexity, making this option attractive for high-throughput applications.Second, in a rate-adaptive scheme based on MN codes, the block length can be kept constant, allowing the introduction of periodic synchronization markers, which has a beneficial effect on frame synchronization performance. Despite these appealing properties, little attention has been paid to MN codes as a means of achieving rate flexibility. A notable exception is the probabilistic amplitude shaping (PAS) scheme presented in

[12] , which proposes a construction reminiscent of MN codes. In PAS, the sequence output from a uniform (binary) source is also processed by a nonlinear block encoder called a distribution matcher (DM) (see

[12] ), which generates a sequence of amplitude symbols with a certain empirical distribution.The binary labels associated with the amplitude symbols are encoded using a non-systematic LDPC code encoder, which generates a parity bit vector. The amplitude symbols, along with the parity bits, are then mapped to pulse amplitude modulation (PAM) symbols. While the main result of PAS is to achieve considerable shaping gains, it was quickly realized that, as a byproduct, PAS enables natural rate matching thanks to the ability to tune the DM rate

[12] , as originally conjectured in [9]. This aspect makes PAS attractive for long-haul optical transmission systems

[13] .

[0033] The present invention preferably proposes a class of rate-adaptive MN codes for the binary-input channel with additive white Gaussian noise (biAWGN). The class is based on protographic LDPC codes

[14] , preferably in conjunction with a Constant Composition Distribution Matcher (CCDM, see

[12] and

[15] ). Since the concatenation of the CCDM with the inner linear block code results in a nonlinear code, an equivalent decoding problem is introduced. The equivalent problem relates to investigating the performance of the protographic LDPC code over the communication channel, where side information is provided to the decoder by observing the LDPC encoder input over a binary-input and binary-output channel.Preferably with the aid of PEXIT analysis (Protograph Extrinsic Information Transfer, see

[16] and

[17] ), rate-adaptive MN codes can be generated within the scope of the present invention, which are capable of operating over a wide range of code rates close to the biAWGN channel capacity.

[0034] Some preferred embodiments of the present invention are explained below. Protograph LDPC codes coupled with a CCDM

[0035] For example, an MN code can be created by concatenating an outer nonlinear code C O with an inner code C I [9]. The present invention particularly considers a special case of rate-adaptive MN codes. In particular, the inner code can be based on protographic LDPC codes coupled with an outer CCDM. More precisely, in this case, the Fig. 4. Here, an independent and identically distributed (also called independent and identically distributed or iid) uniform source generates a message µ ∈ {1, 2, ... , M}. The message can be input into the encoder of an outer code C O of length h, whose task is to generate an output sequence with a given empirical distribution. Following

[12] , such a device is called a DM. The focus here will be on DMs based on constant composition codes (also called constant composition codes or CC codes), which allow a low-complexity implementation using arithmetic coding (see [9] and

[15] ). With ω ∈ (0, 1), let υ be the fractional Hamming weight of the h-bit vector, ie, ω = w H (υ) / h. In addition, the following applies: M=|Co|=(hωh). Consequently, the rate of the outer code Ro=h−1log2M=h−1log2(hωh), which for large h against H b (ω) converges. The output of the DM can then be fed into the encoder of an inner (n, h) binary linear block code C I The inner code C I is defined by CI={c|cH2τ=υH1τ,υ∈F2h} where H1 is an n × h sparse binary matrix and H2 is an n × n sparse binary matrix. Note that, strictly speaking, C I cannot be an LDPC code, ie the code cannot have a sparse control matrix. Nevertheless, C can be regarded as the code obtained by puncturing an (n + h, h) LDPC code with an n × (h + n) control matrix. H=[H1|H2] is determined, where the puncturing is applied to the first h coordinates. Furthermore, let the (n+h, h) LDPC code with a control matrix of the form (2) be considered as the (inner) mother code C IMThe rate of the inner code is R1 = h / n, while the rate of the inner mother code is R IM = h / (n + h) = R I (1 + R I ). It can be assumed that H2 has rank n, which means that H2 is invertible. With G = H1H2 -T be the generator matrix of the inner code C I Note that if the inverse of H2 is dense, the generator matrix G is also dense. It follows that the concatenation of the outer CC code with the inner linear block code results in a marginal distribution of the codeword bits that is approximately uniform, as required by the capacity-achieving input distribution of the biAWGN channel

[12] . The total code resulting from the concatenation of the outer CC code and the inner binary linear block code can be denoted by C, while its rate R = R O R1 ≈ H b (ω)R IIt can be noted that all rates 0 < R < R1 can be achieved simply by fixing the DM parameter and thus its rate, without making any modification (e.g., puncturing / truncation) of the inner code.

[0036] In addition, MN codes based on a Protograph mother LDPC code C IM In particular, the control matrix (2) of the mother code is obtained by lifting a protograph whose base matrix n0 × (h0 + n0) has the form B = [B1 | B2], where B1 has dimensions n0 × h0 and B2 has dimensions n0 × n0, where n0 = n / ℓ and h0 = h / ℓ are positive integers and ℓ is the integer lifting factor. Note that all VNs of type i are punctured with i = 1, ... , h0. An MN code is completely defined by the parity check matrix of the inner mother code C IMand by the DM parameter ω. An MN-code family denotes the set of MN-codes with a fixed mother code that are obtained for all ω ∈ (0, 1) that yield an integer ωh. Similarly, a (protographic) MN-code ensemble Cω(P) the set of MN-codes whose bipartite graph G of the inner mother code is obtained by lifting P and where the DM parameter is ω. The MN-code ensemble family {Cω(P)}ω∈(0,1). is the set of ensembles {Cω(P)}ω∈(0,1). Equivalent parallel channel model

[0037] The analysis of the system described above can pose some challenges in practice. For example, since the C code is nonlinear, the analysis of the density evolution (DE) depends on the transmitted codeword. This makes it difficult to use a reference codeword for the analysis. This problem can be preferably circumvented by resorting to an alternative communication model, which is described in the Fig. 6 and which is equivalent to the one shown in the Fig. 4 (see

[11] and

[18] ). The model is shown in the Fig. 6 is referred to as the equivalent parallel channel (EPC) model, which allows the use of DE analysis for code design. The analysis has some similarities with the DE of LDPC code ensembles developed for joint coding of source and channel

[11] . In the work underlying the present invention, the analysis was performed both with quantized DE (see

[19] ) and with the Gaussian approximation in the form of PEXIT analysis (see

[16] and

[17] ). In both cases, with reference to the EPC model, the a priori channel (which introduces a constant number of errors ωh in z) was replaced by a binary symmetric channel (also called binary symmetric channel or BSC) with crossover probability ω. This choice is justified by the observation that the fraction of errors introduced by the BSC concentrates around ω as h (and n) become large.The following describes, by way of example, how the PEXIT analysis can be performed based on

[17] . However, as stated above, the present invention is explicitly not limited to the PEXIT analysis. With C. BSC (ω) let the capacity of a BSC be denoted by the crossover probability ω and denoted by C AWGN (E s / N0) the capacity of a biAWGN channel with SNR E s / N0. The PEXIT analysis is initialized by setting the mutual information (also called mutual information or MI) at the input of the VNs of type i, where i = 1,...,h0, to C BSC (ω), while for i = h0 + 1, ..., h0 + n0 the MI at the input of the VN of type i is set to C AWGN (E s / N0). The analysis can then be performed using the recursions provided in

[17] and allows the determination (for fixed ω) of the iterative decoding threshold over the biAWGN channel, i.e., the minimum value E s / N0 , for which the MI values ​​tracked by the PEXIT analysis converge to 1. In addition, the threshold is denoted as γ*(R), where the dependence on R (and thus on ω) should be emphasized. Due to the faster computations that PEXIT analysis entails compared to quantized DE, PEXIT analysis can be used preferentially for the optimization of protograms. The threshold γ* can be used as a target value that can be minimized when designing protographs. In the following, it is assumed that a protograph for an inner mother code is to be found that can operate close to the channel capacity over a range [R, R] of rates R, i.e. over a range [ω, ω] of ω values. For this purpose, the parameters of the protograph h0, n0 can be fixed. In this way, the inner code rate RI can be determined. A set of target rates R ⊂ [R, R] can be selected. For each target rate R ∈ can then be the rate of the DM as R O = R / R I which yields the DM parameter ω. Starting from a protograph P The worst-case loss (also called worst-case loss or WCL) can be defined as follows: Δ(P):=maxR∈R[γ*(R)−CAWGN−1(R)]

[0038] The above expression describes the maximum distance between the iterative decoding threshold of the protograph and the biAWGN Shannon bound for the rates in R. Then, the protograph with parameters h0, n0 can be searched that minimizes the WCL in (3). Optimized protocol-based MN codes

[0039] The optimization algorithm described above leads to good protographs, i.e., protographs that are robust for a wide range of code rates. In the following, some application examples for the design of protograph-based MN code ensemble families covering various rate ranges are presented. In all the following examples, according to some preferred embodiments of the invention described above, the search space was limited by setting the maximum number of parallel edges between VN-check node (CN) pairs of the protograph (i.e., the value of the elements of the basis matrix) to 3. In all the following examples, a specific structure of the protograph is considered. The first row is set such that the minimum distance of the code grows linearly. For this purpose, the first h0 + 1 entries must be equal to 3. As already mentioned, the information part (corresponding to the first h0 columns) is punctured.Therefore, a triangular matrix of dimension h0 × h0 is required in the information part to recover the corresponding variable node bits. All other entries are instead subject to the optimization algorithm described above. Example 1:

[0040] Consider a code rate range of [0.1, 0.5]. An MN code family covering this range can be derived from an inner R1 = 1 / 2 code. We search for protographs with 6 VNs and 4 CNs, minimizing the WCL over R = {0.1, 0.3, 0.5}. This results in the following basis matrix: B1 / 2=[33011001|3000210030102001] where the first two columns are assigned to the dotted VNs. Example 2:

[0041] Next, consider a code rate range of [0,1, 0,666]. An MN code family covering this range can be constructed from an inner R I= 2 / 3 code. Protographs with 5 VNs and 3 CNs are searched for, minimizing the WCL over R = {0.1, 0.3, 0.666}. This results in the following basis matrix: B2 / 3=[330110|300310201] where the first two columns are assigned to the dotted VNs. Example 3:

[0042] Finally, consider a code rate range of [0,1, 0,75]. An MN code family covering this range can be constructed from an inner R I = 3 / 4 code. Accordingly, the basis matrix is ​​a 4×7 matrix. Protographs are searched for, minimizing the WCL over R = {0.3, 0.55, 0.75}. This yields the following basis matrix: B3 / 4=[333100010001|3000211031003001] where the VNs assigned to the first three columns are crossed out (also called crossed-out columns or punctured columns).

[0043] The present invention will be explained in more detail below with reference to the figures. Fig. 1 an embodiment of the method according to the invention, Fig. 2 an embodiment of the method steps relating to determining the base matrix, Fig. 3 an embodiment of the system according to the invention comprising a transmitting unit and a receiving unit, Fig. 4 a system model in which an MC code is used to transmit a message over a communication channel, Fig. 5 a modified system model in which an iid scrambler is additionally introduced, Fig. 6 an equivalent system model, Fig. 7 the frame error rate as a function of the SNR for the basis matrix B 1 / 2 , Fig. 8 the frame error rate as a function of the SNR for the basis matrix B 2 / 3 , and Fig. 9 the frame error rate as a function of the SNR for the basis matrix B 3 / 4 .

[0044] In the Fig. 1 schematically illustrates an embodiment of the inventive method 100 for transmitting a message from a transmitting unit to a receiving unit via a communication channel. The method 100 comprises a first method step 110 in which a basis matrix is ​​determined. The basis matrix represents a protograph that can be used to generate an LDPC code. In a second method step 120, an LDPC code is determined using the previously determined basis matrix and the MacKay-Neal code generation method. In a third method step 130, the message to be transmitted is encoded using the previously determined LDPC code. Subsequently, in a fourth method step 140, the encoded message is transmitted from the transmitting unit to the receiving unit via the shared communication channel.

[0045] In the Fig. Figure 2 schematically illustrates an embodiment of the method steps relating to determining 110 the base matrix. In a first sub-step 112, a value range for a desired code rate (also referred to as target code rate) is defined. This value range is defined by a lower range value R1 and an upper range value R2. In a second sub-step 114, n BK Base matrix candidates are determined, where n BK ≥ 2 and preferably greater than 10 or 100. In a third sub-step 116, the quality for each of the n BK Basis matrix candidates are calculated for at least two code rates within the previously defined value range. As already explained above, different analysis methods can be used to evaluate the quality of a basis matrix candidate. In a fourth sub-step 118, the n BKBasis matrix candidates, the basis matrix candidate with the highest quality is selected.

[0046] In the Fig. 3, an embodiment of the system 30 according to the invention is shown schematically. The system 30 according to the invention comprises a transmitting unit 10 and a receiving unit 20. The transmitting unit 10 has a memory unit 12, a computing unit 14 and a communication unit 16. The receiving unit 20 also has a memory unit 22, a computing unit 24 and a communication unit 26. For example, it can be provided that the computing unit 14 of the transmitting unit 10 is designed to determine the basis matrix according to the present invention, to generate an LDPC code from this basis matrix and then to encode a message to be transmitted using the LDPC code. The encoded message can then be transmitted to the transmitting unit 20. Alternatively, it can also be provided that a separate computing unit (in the Fig. 3 not shown) which is configured to generate the base matrix and / or the LDPC code and to provide it to the transmitting unit 10.

[0047] In the Fig. Figure 4 schematically shows a system model in which an MN code is used to transmit a message over a communication channel. As already explained above, an independent source generates a message µ ∈ {1, 2, ... , M}. This message can be input to the encoder of an outer code C0. Such a device can also be referred to as a distribution matcher (DM). In particular, a constant composition distribution matcher (CCDM) can be used. The output of the DM can then be input to the encoder of an inner block code C0. The overall code resulting from the concatenation of the outer code and the inner code can be denoted by C. The modeled codeword x comprising n bits is output at the output of the encoder C. This codeword is transmitted to the receiving unit over the communication channel.At the receiver end, a codeword y containing a noisy signal, which also comprises n bits, is received. This codeword is then decoded by a decoder, which subsequently undergoes a de-matching process.

[0048] In the Fig. Figure 5 schematically shows a modified system model in which an iid scrambler is additionally introduced. The iid scrambler adds a vector z to the input of the inner encoder, which is also provided to the receiving unit. By exploiting the linearity of the transmission channel, the information of the vector z can be used in such a way that adding the vector z does not change the content of the message received by the receiving unit.

[0049] In the Fig. An equivalent system model is shown schematically in Figure 6. This model behaves equivalently to the one in the Fig. 4 shown model. The one in the Fig. The model shown in Figure 6 is also referred to as the equivalent parallel channel model (EPC model) and enables DE analysis for code design. As already explained above, within the scope of the present invention, analyses of the determined codes were performed using DE analysis and also using PEXIT analysis.

[0050] In the Fig. 7-9 are the frame error rates as a function of the signal-to-noise ratio for the basic matrices B 1 / 2 , B 2 / 3 and B 3 / 4These figures show the numerical results on the performance of protograph MN codes over the biAWGN. The results were obtained by Monte Carlo simulations using belief propagation decoding (BP decoding) with full precision and a maximum of 100 iterations. All codes used in the simulations are obtained by lifting the inner LDPC code protograph using a circulant version of the progressive edge growth or PEG algorithm (see

[20] ). The performance of the MN codes constructed from the MN code ensembles of Examples 1, 2, and 3 for different code rates R are shown in the Fig. 7, Fig. 8 and Fig. 9. In the Fig. 7, MN codes of length 12,000 with an inner mother code base matrix were used, while in the Fig. 8 MN codes of length 9,000 with an inner mother code base matrix B 2 / 3 were used and in the Fig. 9 MN codes of length 8,040 with an inner mother code basis matrix B 3 / 4 were used. Remarkably, the diagrams show that very good results can be achieved with a single fixed LDPC code at various rates. LIST OF REFERENCE SYMBOLS 10 transmitter unit 12 Storage unit of the transmitter unit 14 Computing unit of the transmitting unit 16 Communication unit of the transmitting unit 20 receiving unit 22 Storage unit of the receiving unit 24 Computing unit of the receiving unit 26 Communication unit of the receiving unit 30 Communication system 100 procedures 110 first procedural step 112 first sub-step 114 second sub-step 116 third sub-step 118 fourth sub-step 120 second process step 130 third procedural step 140 fourth procedural step LIST OF ABBREVIATIONS LDPC Low-Density Parity Check PAM Pulse Amplitude Modulation BP Belief propagation DE density evolution BPSK binary phase shift keying BSC binary symmetric channel DM distribution matcher iid independent and identically distributed CC constant composition LEO low Earth orbit biAWGN additive white Gaussian noise with binary input (binary-input additive white Gaussian noise) SNR signal-to-noise ratio PEXIT protograph extrinsic information transfer VN variable node CN check node FER frame error rate MN MacKay-Neal EPC equivalent parallel channel WCL worst-case loss PEG progressive edge growth PAS probabilistic amplitude shaping CCDM Constant Composition Distribution Matcher MI mutual information BIBLIOGRAPHY [1] R. Gallager, “Low-density parity-check codes,” Ph.D. Dissertation, Massachusetts Institute of Technology, Cambridge, MA, USA, 1963. [2] T. Richardson und R. Urbanke, Modern coding theory. Cambridge University Press, 2008. [3] A. Morello und V. Mignone, „DVB-S2: The second generation standard for satellite broad-band services“, Proc. IEEE, vol. 94, no. 1, pp. 210-227, Jan. 2006. [4] T. Tian und C. R. Jones, „Construction of rate-compatible Idpc codes utilizing information shortening and parity puncturing,“ EURASIP Journal on wireless communications and networking, vol. 2005, no. 5, S. 1-7, 2005. [5] S. Dolinar, „A rate-compatible family of protograph-based Idpc codes built by expurgation and lengthening,“ in Proceedings. Internationales Symposium über Informationstheorie, 2005. ISIT 2005. IEEE, 2005, S. 1627-1631. [6] T. Van Nguyen, A. Nosratinia, and D. Divsalar, „The design of rate-compatible protograph Idpc codes,“ IEEE Transactions on communications, vol. 60, no. 10, pp. 2841-2850, 2012. [7] T.-Y. Chen, K. Vakilinia, D. Divsalar, and R. D. Wesel, „Protograph-based raptor-like Idpc codes,“ IEEE Transactions on Communications, vol. 63, no. 5, pp. 1522-1532, 2015. [8] T. Richardson und S. Kudekar, „Design of low-density parity check codes for 5g new radio,“ IEEE Communications Magazine, vol. 56, no. 3, pp. 28-34, 2018. [9] D. MacKay, „Good error-correcting codes based on very sparse matrices,“ IEEE Trans. Inf. Theory, vol. 45, no. 2, pp. 399-431, Mar. 1999.

[10] M. J. Wainwright und E. Martinian, „Low-density graph codes that are optimal for binning and coding with side information,“ IEEE Trans. Inf. Theory, vol. 55, no. 3, pp. 1061-1079, Mar. 2009.

[11] M. Fresia, F. Perez-Cruz, H. V. Poor, and S. Verdu, „Joint source and channel coding,“ IEEE Signal Process. Mag., Bd. 27, Nr. 6, S. 104-113, Nov. 2010.

[12] G. Böcherer, F. Steiner, and P. Schulte, „Bandwidth efficient and rate-matched low-density parity-check coded modulation,“ IEEE Trans. Commun, vol. 63, no. 12, pp. 4651-4665, Dec. 2015.

[13] F. Buchali, F. Steiner, G. Böcherer, L. Schmalen, P. Schulte, and W. Idler, „Rate adaptation and reach increase by probabilistically shaped 64-QAM: An experimental demonstration,“ J. Lightw. Technol., vol. 34, no. 7, pp. 1599-1609, Apr. 2016.

[14] J. Thorpe, „Low-density parity-check (LDPC) codes constructed from protographs“, NASA JPL, IPN Progr. Rep. 42-154, Aug. 2003.

[15] P. Schulte und G. Böcherer, „Constant composition distribution matching,“ IEEE Trans. Inf. Theory, vol. 62, no. 1, pp. 430-434, Jan. 2016.

[16] S. ten Brink, „Convergence behavior of iteratively decoded parallel concatenated codes,“ IEEE Trans. Commun, vol. 49, no. 10, pp. 1727-1737, Oct. 2001.

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Claims

[1] Method (100) for transmitting a message from a transmitting unit (10) to a receiving unit (20) via a communication channel, the method (100) comprising the following steps: - determining (110) a basis matrix representing a protograph; - determining (120) an LDPC code using the previously determined basis matrix and the MacKay-Neal code generation method; - coding (130) the message to be transmitted using the previously determined LDPC code; and - transmission (140) of the coded message from the transmitting unit (10) to the receiving unit (20); - wherein determining (110) the base matrix comprises: - defining (112) a value range for a target code rate, the value range being defined by a lower range value R1 and an upper range value R2; - Determine (114) of n BK Basis matrix candidates, where nBK ≥ 2; - Calculate (116) the quality for each of the n BK Basis matrix candidates for at least two code rates each within the previously defined value range; - Selecting (118) the basis matrix candidate with the highest quality from the n BK Base matrix candidates; and - Set the base matrix to the following value: B1 / 2=[33011001|3000210030102001]. [2] Method (100) according to claim 1, characterized by that the method (100) is used in a satellite communication system. [3] Method (100) according to one of claims 1 to 2, characterized by that determining (120) an LDPC code using the previously determined basis matrix and the MacKay-Neal code generation method comprises concatenating an inner code and an outer code. [4] Method (100) according to one of claims 1 to 3, characterized by the following procedural steps: - Measurement of channel quality over a specified measurement period, - Specifying the value range for the target code rate depending on the channel quality determined during the measurement period. [5] Communication system (30) comprising a transmitting unit (10), a receiving unit (20) and a computing unit (14), wherein the computing unit (14) is designed to - to determine a basis matrix that represents a protograph; - to determine an LDPC code using the previously determined basis matrix and the MacKay-Neal code generation method; - where determining the base matrix comprises: - Setting a range for a desired code rate, the range being defined by a lower range value R1 and an upper range value R2; - Determine n BK Basis matrix candidates, where n BK ≥ 2; - Calculate the quality of the n BKBasis matrix candidates for at least two code rates within the previously specified range; - Selecting the basis matrix candidate with the highest quality from the n BK Base matrix candidates; and - Set the base matrix to the following value: B1 / 2=[33011001|3000210030102001] and wherein the transmitting unit (10) is designed to - encode the message using the previously determined LDPC code; and - to transmit the coded message to the receiving unit (20).

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