Communication facilities and methods for message identification

Abstract

A communication device may include: a processor configured to perform a procedure comprising: receiving a first consensus check code from a first communication device; receiving a second consensus check code from a second communication device; generating a combination code by network encoding the first consensus check code and the second consensus check code; and instructing the transmission of the combination code to the first communication device and to the second communication device.

Description

The invention relates to communication devices and methods. In a conventional communication protocol without network coding, a communication system requires four time slots to transmit two data packets between two communication devices ("Alice" and "Bob") via a relay communication device ("Rudy") and to fulfill the communication task. - First step: "Alice" sends a data packet to a relay communication device (hereinafter also referred to as "Rudy"). - Second step: The relay communication device forwards the data packet to "Bob". - Third step: Bob sends a data packet to the relay communication device. - Fourth step: The relay communication device forwards the data packet to Alice. XU, Hong; LI, Baochun: Network coding in relay-based networks (Cooperative cellular wireless networks. Cambridge, Cambridge University Press, 2011. pp. 277-298. - ISBN 9780521767125 ) describes network coding at the physical level and its application in relay-based networks. POPOVSKI, Petar; YOMO, Hiroyuki: Physical network coding in two-way wireless relay channels (Proceedings of 2007 IEEE International Conference on Communications (ICC), 24-28 June 2007, Glasgow, United Kingdom, pp. 707-712. - ISBN 1-4244-0353-7) describes an application of network coding for two-way relay channels, where two nodes communicate with each other using a third relay node. KOIKE-AKINO, Tsohiaki [et al.]: Non-linear network coding in two-way relaying discrete channels (2009 international conference on wireless communications & signal processing, 13-15 November 2009, Nanjing China. IEEE, 2009. 5 pp. ISBN 978-1-4244-5668-0) describes a network coding in which there are two communicating nodes and one relay node. Exemplary embodiments of the invention are shown in the figures and are explained in more detail below. Figure 1 shows a communication protocol without network coding; Figures 2A to 2D show two approaches with identification codes (consensus checking) and with traditional linear network coding, each compared with message transmission, where Figures 2A and 2B show linear approaches and Figures 2C and 2D show non-linear approaches; Figure 3 shows a communication system according to various aspects of this disclosure; Figure 4 shows a communication system according to various aspects of this disclosure; Figure 5 shows a method according to various aspects of this disclosure; and Figure 6 shows a method according to various aspects of this disclosure. The following detailed description refers to the accompanying drawings, which form part thereof and illustrate specific embodiments in which the invention can be implemented. In this context, directional terminology such as "top," "bottom," "front," "back," "anterior," "rear," etc., is used with reference to the orientation of the described figure(s). Since components of embodiments can be positioned in a number of different orientations, the directional terminology serves only for illustration and is in no way limiting. It is understood that other embodiments may be used and structural or logical modifications may be made without deviating from the scope of protection of the present invention.It is understood that the features of the various exemplary embodiments described herein can be combined with one another, unless specifically stated otherwise. The following detailed description is therefore not to be interpreted in a limiting sense, and the scope of protection of the present invention is defined by the appended claims. Within the scope of this description, the terms "connected," "attached," and "coupled" are used to describe both direct and indirect connections, direct or indirect links, and direct or indirect couplings. In the figures, identical or similar elements are labeled with identical reference symbols where appropriate. Within the scope of this description, a (genuine) consensus verification code can be understood as the result of a mapping using a function that assigns a string (in other words, a codeword) to each message, so that the codewords can be sent over a noisy channel and, based on the received, i.e.,noisy, version of the codewords, with a sufficiently high probability (for example, with a probability of at least 0.95, for example, with a probability of at least 0.97, for example, with a probability of at least 0.98, for example, with a probability of at least 0.99, for example, with a probability of at least 0.995, for example, with a probability of at least 0.999 or higher) it can be correctly estimated whether the message is the same as any other message, without the codeword being transmitted / decoded in plaintext or without the receiver needing to be able to decode the codeword in plaintext. Such codes can be constructed, for example, from deterministic identification codes by encoding the messages of both end nodes (e.g., a first communication device "Alice" as the first end node and a second communication device "Bob" as the second end node) with an identification code (ID code). Identification is similar to consensus checking, except that in this case, only Alice sends a message directly to Bob, without a relay communication device, while the other message is known only to Bob. Ultimately, only Bob should know whether Alice's message and his message are the same. Generally, a good ID code is a one-to-one mapping to an (e.g., linear) error-correcting code that has a positive relative distance (e.g., Hamming distance).For example, the message can be expressed as a vector in a vector space over a finite field and multiplied by the generator matrix of a linear code (c = G · x, where c is the codeword and x is the message word, and G is the generator matrix of the code). The resulting vector, or the corresponding string, is then a codeword. A conventional construction for deterministic ID codes uses a concatenated Reed-Solomon code as a linear code consisting of an inner and an outer code: The alphabet of the inner code, for any positive block length n, any positive b, and a < b < 2a, must be a prime power. Furthermore, the inner code has any length n1, 0 < n1 < n, and a minimum Hamming distance (i.e., the number of symbols by which two strings differ) d1 ≥ ε1·n1. Thus, the inner code has the size q2 = M1, and the outer code has the alphabet size q2 = M1, length d2 = ε2 · n2, and size d2 = ε2 · n2. The concatenated code therefore has alphabet size q1, length n, distance d ≥ ε1ε2n, and size For example, one can encode by first encoding the message with the outer code, and then encoding each symbol of the resulting string with the inner code. Within the scope of this description, a (genuine) consensus check code can also be understood as the result of a mapping using a function that assigns a string (in other words, a codeword) to each message, such that the codewords can be sent over a noisy channel and, based on the received, i.e., noisy, version of the codewords, it can be correctly estimated with a sufficiently high probability (for example, with a probability of at least 0.95, for example, with a probability of at least 0.97, for example, with a probability of at least 0.98, for example, with a probability of at least 0.99, for example, with a probability of at least 0.995, for example, with a probability of at least 0.999 or higher) whether the message is the same as any other message.Additionally, the ratio of the number of messages to the number of codewords must be greater than with a single code for message transmission. Various aspects of this disclosure provide a communication protocol for verifying consensus between two communication partners (communication entities, also referred to as parties) who can only communicate via a relay. A relay may include a relay communication device. A relay communication device may include one or more communication devices, which may, for example, form a communication network node or be distributed throughout a communication network. Intuitively, one or more novel post-Shannon identification codes are used to reduce the amount of information required to reach consensus. Intuitively, a protocol of (linear or non-linear) network coding for post-Shannon communication procedures is provided. Several aspects of this revelation are based on a conventional network coding for relay-based communication. In this communication approach, two communication devices (hereinafter referred to for simplicity as "Alice" (otherwise first communication device 102) and "Bob" (otherwise second communication device 106) want to exchange information packets. If the conventional digital network coding of a relay communication device 104 (also referred to as "Rudy") allows mathematical operations to be performed on the received data packets, the transmission of two data packets between two communication devices ("Alice" and "Bob") via a relay communication device ("Rudy") can also be achieved with only three time slots: - First step: Alice 102 sends a data packet to the relay communication device 104. - Second step: Bob 106 sends a data packet to the relay communication device 104. - Third step: The relay communication device 104 combines both received data packets, e.g. using an XOR (exclusive OR) operation, and sends the XORed data packet to Alice 102 and Bob 106.Since Alice 102 and Bob 106 have information about their sent data packets, they can each decrypt the other's data packet by XORing their own data packet with the data packet received from the relay communication device 104. With conventional analog (or physical) network coding, the number of transmissions can be further reduced. If perfect synchronization exists between three communication network nodes (e.g., Alice 102, Rudy 104, and Bob 106), Alice 102 and Bob 106 can, in a first step, simultaneously send their messages (i.e., data packets) to a relay communication device 104 ("Rudy") (references 108 and 112). In a second step, the relay communication device 104 receives the superposition of the transmitted signals, allowing it to encode and transmit a data packet (references 110, 114) that is a function of the received signal. This scheme requires only two time slots to carry out the transmission. This approach is illustrated in Fig. 1. The so-called message identification is a post-Shannon (goal-oriented) communication scheme in which the goal of the communication is different from that of traditional message transmission. In conventional message transmission, a number of events can occur at the sender's end. When an event occurs, the sender encodes it into a message and sends it to a receiver. The receiver decodes (e.g., decrypts) the message and responds figuratively: "What event occurred at the sender's end?" Although only one answer is correct, there are as many possible answers to this question as there are possible events at the sender's end. When identifying messages, the goal is different. The recipient is only interested in an event. The recipient doesn't care which event occurred at the sender's end; they only want to know if it was the event they're interested in or not. Therefore, the recipient wants to answer the question: "Did the event I'm interested in occur at the sender's end?" Unlike message transmission, message identification has only two possible answers: "Yes" or "No." For example, imagine a recipient who plays the lottery. They aren't interested in knowing the exact number the sender chose. They only want to know if their lottery ticket won or not. It has been proven that the sender needs to transmit significantly fewer bits if the goal of the communication is (merely) message identification. This result is interesting because the same goal can be achieved with fewer bits. Shannon proved that, for a given channel capacity C, using n bits, 2nC events can be encoded for message transmission. With identification codes, without using additional resources such as random numbers, approximately 2n events can be encoded to enable correct identification over channels with a binary input alphabet. Thus, the same rate is achieved that would only be attainable for message transmission on a perfect, noise-free channel. This is possible because the receiver's goal is not to decode the original message (e.g., to retrieve the original message).to decode the message), but rather to determine whether the received message matches the message the receiver is interested in. In other words, the receiver wants to verify the match with the sent message. For Gaussian channels and with a power limitation, a genuine identification code can reliably process messages using n channel symbols, whereas a transmission code can only process 2n log(1+SNR) messages, where SNR denotes the ratio of signal power to noise power, i.e., the signal-to-noise ratio. Several aspects of this disclosure have potential advantages of targeted network coding at the physical layer (also referred to as the physical layer, for example, according to the so-called OSI model) for message identification, i.e., for consensus checking, in a multi-hop communication network. Several aspects of this disclosure describe a communication network with three communication network nodes, two of which can be "edge nodes," representing users Alice 102 and Bob 106, as illustrated in Figure 1, who can only communicate with the help of a relay node, Rudy (i.e., the relay communication device 104) (see Figure 1). Several aspects of this disclosure concern the problem of consensus checking (in other words, message identification) between two communication devices, for example, two "edge nodes," i.e.,Both communication network nodes (i.e., both communication facilities) select a message and want to find out if the message they selected matches that of the other communication network node. Various aspects of this revelation are used, for example, for robot swarms or digital twin applications, where multiple robots or the real (physical) robots and their digital twins want to know if there is a consensus for the next decisions to be made. Various applications typically have more than two communication network nodes, and several aspects of this revelation can readily be extended to K-user relay communication networks for each K, independent of the block length. Several aspects extend this setup beyond message transmission and linear recoding to non-linear network codes, demonstrating the advantages of goal-oriented, physical-layer network coding for this particular communication problem of message identification. In several aspects, the communication network nodes use a novel type of deterministic ID code to encode their message for the relay communication device. Codewords are sent over noisy channels to the relay communication device, which statistically checks if the messages are identical and sends a bit, the test result.It is shown that the capacity for such network codes is generally limited above and below by the ID capacities of certain modified uplink channels (as will be explained in more detail below). Thus, the capacity is determined for pairs of q-ary symmetric and erase channels, as well as for binary adder channels in the uplink, which the code approximates according to various aspects of this revelation. For K many users, the advantages of the optimal non-linear network coding revealed here in Figures 2C and 2D are even greater: With linear network coding, the protocol would either be limited to two users, or it would take considerably more time, namely presumably K-1 times as long as for only two users. However, since in the non-linear case only one bit is transmitted over the downlink, regardless of the number of users, the rate is not reduced, nor is the time and transmission resource requirement increased. Figures 2A and 2B show two approaches: one using identification codes (in other words, consensus verification codes) and the other using traditional linear network coding, each compared to the message transmission described above. Figures 2A and 2B show linear approaches, and Figures 2C and 2D show non-linear approaches. The length of the arrows in the figures describes the relative length of the messages that need to be transmitted to achieve consensus checking. Two parameters can be changed in a more intuitive way: transmission codes (T-codes, abbreviated as T in Fig. 2A) vs. consensus verification codes (CT-codes, abbreviated as CT in Fig. 2B); or linear network coding vs. non-linear network coding. There are therefore four possible scenarios (shown in Fig. 2A to 2D). The duration of upstream and downstream connections for various network codes is clearly compared when using multiple-access transmit codes (labeled T in Figures 2A to 2D) or multiple-access consensus check codes (labeled CT in Figures 2A to 2D). The "+" sign denotes letter-by-letter addition of sequences, e.g., an XOR operation. A transmission of exactly one bit is denoted by the reference symbol T1. First, linear network coding is explained (see Fig. 2A and Fig. 2B). In this case, there are two possibilities. In a first scenario 200, linear network coding and conventional message transmission are described (see Fig. 2A). This case is the same as the analog linear network coding described above in connection with Fig. 1. In a second scenario 210 (see Fig. 2B), the system performance using linear network coding and identification codes is described. Alice 102 and Bob 106 encode their respective events using an identification code (also called a consensus check code) CT. The relay communication device 104 uses linear network coding, as described above in conjunction with Fig. 1, to relay this information to Alice 102 and Bob 106. The message lengths are shorter than in the first scenario 200 above because the CT codes allow fewer bits to be used for consensus checking, as described above for identification codes. In a third scenario 220 (see Fig. 2C), a first non-linear network coding scenario is described. In this third scenario 220, Alice 102 and Bob 106 use transmission codes to inform the relay communication device 104 of their status. The relay communication device 104 performs a consensus check, e.g., whether the messages are identical or not. Then, the relay communication device 104 sends (exactly) one bit T1 to the communication network nodes (i.e., to the first communication device (Alice) 102 and the second communication device (Bob) 106), indicating whether a consensus exists or not. The consensus check at the relay communication device 104 is a non-linear process. Therefore, this third scenario 220 is also referred to as non-linear network coding.In the third scenario, a transmission of long messages T for the transmission codes is provided first, followed by a (one-bit) transmission from the relay communication device 104. In a fourth scenario 230 (see Fig. 2D), CT codes are used in combination with non-linear network coding. In this fourth scenario 230, Alice 102 and Bob 106 encode their information as CT codes CT and transmit them as corresponding data packets (in other words, messages) CT to the relay communication device 104. The length of the codewords used to test the consensus of these messages is much shorter (see below for a more detailed explanation) than that of the transmission codes T. Since the CT codes CT are sufficient to test consensus, the relay communication device 104 can then send exactly one bit to the communication network nodes (i.e., to the first communication device (Alice) 102 and the second communication device (Bob) 106) to determine whether consensus exists.The communication protocol according to the fourth scenario 230 achieves the goal (consensus verification between Alice 102 and Bob 106) while simultaneously exchanging the smallest possible amount of data. This will be explained in more detail below. The scenarios described above, for example, the second scenario 210 and the fourth scenario 230, can be extended in various aspects of this revelation to scenarios in which K communication network nodes (K is any natural number, for example, in a range from two communication network nodes to 100 or more communication network nodes, for example, in a range from two or three communication network nodes to 80 communication network nodes, for example, in a range from two or three communication network nodes to 70 communication network nodes, for example, in a range from three communication network nodes to 50 communication network nodes, for example, in a range from three communication network nodes to 30 communication network nodes, for example, in a range from four communication network nodes to 25 communication network nodes,For example, in a range of five to 20 communication network nodes, or in a range of six to 10 communication network nodes, a consensus test might be performed. The respective scenario would be similar. First, the K communication network nodes transmit their status, encoded with a CT code, to the relay communication device 104. The relay communication device 104 checks whether a consensus exists and sends the result in the form of a single information bit T1 to the communication network nodes (where the information bit indicates whether a consensus exists or not, in other words, whether the checked messages have a message identity or not). Several aspects of this disclosure are intended, for example, for satellite communications or relay systems in general, as they offer significant rate improvements for consensus testing without requiring substantially more computational effort at relay communication facility 104, and with less computational effort for decoding at Alice 102 and Bob 106, thus reducing overall resource consumption. Alice 102 and Bob 106 (i.e., the first communication facility 102 and the second communication facility 106) perform the encryption and transmission of the encrypted identities, while relay communication facility 104 merely performs the test (i.e., the consensus check) and transmits only exactly one bit T1 (i.e., the information bit T1) to the other communication facilities (for example, via a broadcast channel).Since this transmission is cost-effective, the relay communication device 104 can consume less energy overall. In conventional relay communication scenarios, the broadcast channel (BC) often represented the bottleneck. Its capacity for message transmission remains an open problem, except in special cases such as degraded BCs and Gaussian MIMO BCs. However, since the relay communication device 104 in the communication protocol, according to various aspects of this disclosure, only needs to transmit exactly one bit T1, the consensus-checking capacity of the communication network under consideration is equal to the generalized consensus-checking capacity of the uplink communication link (the task then being for the relay Rudy to estimate whether the two messages from Alice and Bob are the same). Fig. 3 shows a communication system according to the second scenario 210. In this scenario 210, the first communication device 102 has a processor 302, a memory 304, and a transceiver 306. The processor 302, the memory 304, and the transceiver 306 are connected, for example, by means of a data bus 308. A first data packet 310 to be encoded is stored in the memory 304. The processor 302 reads the first data packet 310 to be encoded from the memory 304 and encodes it into a first consensus check code 312 to be transmitted, for example, according to one of the figures described above. The processor 302 sends the first consensus check code 312 to the transmit-receive unit 306. The transmit-receive unit 306 sends the first consensus check code 312 to the relay communication device 104. In this scenario 210, the second communication device 106 also has a processor 314, a memory 316, and a transceiver 318. The processor 314, the memory 316, and the transceiver 318 are connected to each other, for example, by means of a data bus 320. A second data packet 322, to be encoded, is stored in the memory 316. Processor 314 reads the second data packet 322 to be encoded from memory 316 and encodes it into a second consensus check code 324 to be transmitted, for example, according to one of the diagrams described above, such as the same diagram performed by processor 302 of the first communication device 102. Processor 314 sends the second consensus check code 324 to the transmit-receive unit 318. The transmit-receive unit 318 then sends the second consensus check code 324 to the relay communication device 104. The relay communication device 104 comprises a processor 326, a memory 328, and a transceiver 330. The processor 326, the memory 328, and the transceiver 330 are interconnected, for example, by means of a data bus 332. The transmit-receive unit 330 receives the first consensus check code 312 and the second consensus check code 324 and stores them in memory 328. The processor 326 reads the first consensus check code 312 and the second consensus check code 324 from memory 328 and executes the procedure described below. In other words, the processor 326 is configured to receive the first consensus check code 312 from the first communication unit 102, to receive the second consensus check code 324 from the second communication unit 106, to generate a linear combination code 334 by linear network encoding of the first consensus check code 312 and the second consensus check code 324, and to instruct the transmission of the linear combination code 334 to the first communication unit 102 and to the second communication unit 106. Thus, the linear combination code 334 is supplied to the transmit-receive unit 330, and the transmit-receive unit 330 sends the linear combination code 334 to the first communication device 102. Furthermore, the transmit-receive unit 330 sends the linear combination code 334 to the second communication device 106. The first communication device 102 receives the linear combination code 334 (for example, by means of the transceiver 306) and performs one or more actions based on the content of the linear combination code 334. The second communication device 106 also receives the linear combination code 334 (for example, via the transceiver 318) and performs one or more actions based on the content of the linear combination code 334. Fig. 4 shows a communication system according to the fourth scenario 230. The communication system according to the fourth scenario 230 is similar to the communication system according to the second scenario 210. Therefore, only the differences between the communication system according to the fourth scenario 230 and the communication system according to the second scenario 210 are described below. For the similar features, please refer to the description of the communication system according to the second scenario 210. The transmit-receive unit 330 of the relay communication device receives the first consensus check code 312 and the second consensus check code 324 and stores them in memory 328. The processor 326 reads the first consensus check code 312 and the second consensus check code 324 from memory 328 and executes the procedure described below. In other words, the processor 326 is configured to receive the first consensus check code 312 from the first communication device 102, to receive the second consensus check code 324 from the second communication device 106, to generate a non-linear combination code (of exactly one bit length) 402 by non-linear network encoding of the first consensus check code 312 and the second consensus check code 324, and to instruct the transmission of the non-linear combination code 402 to the first communication device 102 and to the second communication device 106.The non-linear combination code 402 thus contains only the information whether the first consensus check code 312 matches the second consensus check code 324 (for example, represented by the bit of the combination code 402 with a first bit value (for example, logical "0")) or whether the first consensus check code 312 does not match the second consensus check code 324 (for example, represented by the bit of the combination code 402 with a second bit value (for example, logical "1")), which is different from the first bit value. Thus, the non-linear combination code 402 is supplied to the transmit-receive unit 330, and the transmit-receive unit 330 sends the non-linear combination code 402 to the first communication device 102. Furthermore, the transmit-receive unit 330 sends the non-linear combination code 334 to the second communication device 106. The first communication device 102 receives the non-linear combination code 402 (for example, by means of the transceiver 306) and performs one or more actions based on the content of the non-linear combination code 402. The second communication device 106 also receives the non-linear combination code 402 (for example, via the transceiver 318) and performs one or more actions based on the content of the non-linear combination code 402. It should be noted that the first communication device 102, the second communication device 106, and the relay communication device 104 can each be implemented by means of one or more communication devices with one or more processors. Furthermore, the relay communication device 104 can also be implemented by one or more entities that may be located within the transmission path itself (each of which may have one or more processors). The entities may also be configured to perform the relay communication device procedure jointly (with any division of the respective steps partly within the one or more entities and partly within the relay communication device). Furthermore, it should be noted that in general the respective messages (i.e., for example, the consensus verification codes 312, 324 and / or the combination codes 334, 402) can be transmitted wired and / or wirelessly (for example, via a radio interface) between the communication devices 102, 104, 106. Fig. 5 shows a flowchart illustrating a method 500 according to various aspects of this disclosure. The method may include, in 502, receiving a first consensus check code from a first communication device, in 504, receiving a second consensus check code from a second communication device, in 506, generating a combination code by network encoding the first consensus check code and the second consensus check code, and in 508, instructing the transmission of the combination code to the first communication device and to the second communication device. Fig. 6 shows a flowchart illustrating a method 600 according to various aspects of this disclosure. The method may include, in 602, generating a first consensus check code from a first communication device, in 604, instructing the transmission of the first consensus check code to another communication device, in 606, receiving a combination code formed by network encoding the first consensus check code and a second consensus check code generated by yet another communication device, and in 608, checking the combination code. The following describes further details on consensus verification over relay communication networks through communication network coding at the physical layer. Physical layer network codes for consensus checks over a relay communication facility are disclosed, where each of several communication facilities knows one message and wants to determine if all messages are identical. It is assumed that the encoders cannot randomize. The codes are generalized deterministic identification (ID) codes for the uplink multiple-access channel to the relay communication facility. Using a novel derandomization result for decoders, it is shown that the CT capacity of a relay communication network is generally bounded below by the CT capacity of the uplink communication link alone and above by the ID capacity of modified uplink channels.For uplink channels that are pairs of q-ary symmetric channels or erasure channels (which erase a symbol with a certain probability by replacing the symbol with an erasure marker (for example, a symbol "e") (hereafter also referred to as erasure channels) or binary adder channels, it is shown that the proposed code achieves the CT capacity equivalent to that of ID over a noise-free uplink channel. For pairs of Gaussian channels and Gaussian adder channels, the capacity limits are equal to those of ID over a partial uplink channel, and the code sizes scale superexponentially with block length, similar to ID codes. The asymptotically achievable code sizes and required block lengths are compared with the transmission coding for an uplink communication link and with additive linear network codes, both of which prove to be suboptimal. In its early stages, information theory investigated, on the one hand, problems that were primarily single-hop problems in nature, i.e., sender-receiver pairs communicating over a noisy channel. On the other hand, as explained above, information theory also addressed the technical problem of communication where the communication goal is message transmission, i.e., "the reproduction of symbols selected at one point at another point," regardless of meaning or communication goals. A new class of problems was discovered in which multiple senders and receivers are connected in a multicast communication network, where different communication network nodes have different communication links, thus breaking down the point-to-point nature of communication. Targeted communication systems that consider the communication intent as part of their design have been investigated. This can lead to efficiency gains, as only as many bits are transmitted as are necessary to achieve the goal. For the goal of message identification (ID), for example, exponential data reductions are possible by using stochastic coding; that is, ID codes of a block length n and rate R have a size of 22nR under stochastic coding, compared to a transmission code of size 2nR, where R approaches the capacity. With deterministic coding for continuous-valued channels, ID codes can have a size of... This is possible because the receiver's goal is not to decrypt the original message, but to determine whether it matches the message of their own interest. In other words, the receiver wants to verify the match with the transmitted message. Message ID has been studied with and without stochastic coding for various channel models. However, like the original message transmission, ID has only been studied in single-hop scenarios and with passive feedback. Various aspects of this disclosure describe different potential advantages of targeted network coding at the physical layer for message identification (CT) across multi-hop communication networks. A communication network with three communication network nodes is considered, two of which are edge nodes representing the user communication devices Alice 102 and Bob 106. These can only communicate with the help of a third so-called relay node (the relay communication device 104), Rudy (see Fig. 1). The problem of CT between the two edge nodes is described; that is, each communication network node selects a message and determines whether the message they selected matches that of the other communication network node. CT in networked communication systems is suitable in various aspects for controlling multiple mobile autonomous robots. In a practical application, there will typically be more than two robots, and various aspects can readily be extended to relay communication networks with more than two communication devices. In an Alice-Rudy-Bob communication scenario, as described here, linear network coding has demonstrated efficiency gains in message transmission in several applications, such as satellite communication or molecular communication. This setup extends beyond message transmission and linear recoding to non-linear network codes designed for consensus checking, thus revealing the advantages of receiver-oriented network coding at the physical layer for this particular problem.In various aspects, the communication network nodes use a type of deterministic ID code, also known as a CT code, to encode their message for the relay communication equipment. The codewords are sent to the relay communication equipment over one or more noisy channels, which statistically checks whether the messages are identical and relays the one-bit result back. It can be shown that the capacity for such network codes is limited both above and below by the CT and ID capacities of certain modified uplink channels. Thus, the capacity is determined for networks where the uplink is a pair of q-ary symmetric or erase channels, as well as for binary adder channels as the uplink. This is done using a capacity-reaching code construction, as illustrated in Protocol 230 (see Fig. 2D and Fig. 4) in this description. For consensus checking over pairs of parallel Gaussian channels and over Gaussian adder channels, the same capacity limits and code sizes 2Rnlogn apply as for identification over Gaussian channels. These rates and code sizes are achieved by a construction as described in Protocol 230 (see Fig. 2D and Fig. 4) in this description. For the described noisy discrete channels, the capacities exceed the rates achievable with transmission codes in an uplink communication link. They are at least twice as high as the rates achievable with additive recoding at the relay. For Gaussian channels, codes even grow faster in block length than transmission-based or linear network codes, such as identification codes for channels with uncountable alphabets (e.g., Gaussian channels). A set of probability mass functions (PMFs) over a set is denoted by . A channel is a function where denotes the set of random variables (RVs) over . A multiple-access channel (MAC) has two inputs and one output. A broadcast channel (BC) has one input and two outputs. The sets can be finite (discrete channels) or even uncountable, as in ℝ for Gaussian channels. One focus is on memoryless channels, denoted by Yn, where each symbol of an input sequence xn = [x1,...,xn] is transmitted independently, i.e., the channel output is given by Yn(xn) = [Y(x1),...,Y(xn)]. The identity channel over the set is denoted by . Definition 1: Assume a MAC uplink communication connection: any BC downlink communication connection: any set. A (M,n,l,Up,Dwn, λ,T) consensus check code (CT code) for a relay communication network (Up, Dwn) is a function tuple G = (E1,E2,Rec,D1,D2), where for all m, m ∈ [M], where and satisfy the inputs for the condition, for example, the performance condition. A consensus check code has three phases: 1) Each user k = 1, 2 encodes a message mk ∈ [M] with an encoder and sends it to the relay communication facility via Upn. 2) The relay communication facility encodes its received sequences according to Rec: and sends them around via Dwnl. 3) The users test whether m1 = m2 holds true. Definition 2: Assume an arbitrary MAC and a (M,n,Y,λ,T)-MAC consensus check code (CT code) is a triple (E1, E2, D) of encoders for k = 1,2, and a decoder such that and for all m, m ε [M], A rate can be achieved if, for every λ > 0 and sufficiently large n, there exists a (2s(n)R,n,Y,λ,T)-CT code. The CT capacity C(s,Y,T) is the supremum of all achievable rates. One can also force the use of a transmission code for CT: By comparing the decoded messages, a CT code is obtained. Definition 3: Assume an arbitrary MAC and a (M,n,Y,λ,T) transmission code (MAC-T code) is a triple (E1, E2, D) of encoders for k = 1,2, and a decoder such that for all m, m ∈ [M], and Definition 4 (Transmission via BC with shared messages): Assume a broadcast channel (BC). A (M,l,B,λ) transmission code (BC-T code) is a triple (E, D1, D2) of an encoder and two decoders such that, for any message m ∈ [M], and with high probability, both decoders correctly estimate the message m, i.e. Achievable rates and T-capacities are defined in the same way as for CT. For general channels, the following limits are proven, which make it possible to reduce the consensus verification problem over a relay communication facility to simpler identification problems. Theorem 1: For all MAC Up, BC Dwn, and monotonically non-decreasing scales s, if Dwn has a positive T-capacitance, and for all t > 0, that If Up is a pair of parallel channels (Up = [Up1, Up2]) then: CT for a channel [Y,idℕ], where is the same as identification (ID) for channel Y. Capacity results are obtained for networks with these uplinks: 1) Two parallel q-ary symmetric channels (QSCs): QSCq,γk(x) := x + Zkmod q , for each k = 1,2, where x, Yk and Zk are q-ary, and also PZk(0) = 1 - γk, and for each z ≠ 0, it is assumed that 2) Noisy binary adder channel (q = 2): BACγ1,γ2(x1,x2) := QSC2,γ1(x1) + QSC2,γ2(x2)), where the output Y is ternary. This channel model is used to describe the uplink in a molecular relay communication network with only one sensor in the relay communication device. As above, it is assumed that 3) Two q-ary deletion channels (QECs): Yk(x) = QECηk(x), i.e., Pr{Yk(x) = x} = 1 - ηk and Pr{Yk(x) = deletion} = ηk, where 0 ≤ η1 ≤ η2 < 1. 4) Parallel Gaussian channels: Yk(x) = x + Zk, for each k = 1, 2, where x, Yk, Z are gravitational and Z is normally distributed. 5) Gaussian adder channel: Y(x1, x2) = x1 + x2 + Z, where x1, x2, Y, Z are real and Zk ~ N(0, σ2) is normally distributed. Theorem 2 (Channels 1 to 3): For any BC Dwn with positive T-capacitance, any q > 0, T1⊆ T2⊆ P([q]) and γ1,γ2,η1,η2 and a MAC Up as described for channels 1 to 3 (parallel QSC, QEC, or BAC), the consensus checking capacity is given by in particular It should be noted that the scaling s(n) = n is the normal case for message transmission. In the Gaussian case, however, s(n) = n log n. Theorem 3 (Channels 4 and 5): Dwn is any BC with positive T-capacitance, and Up is given by one of channels 4 or 5, where for power conditions Tk= {PX: E|X2| ≤ Ak} and A1≤ A2, Theorem 4: Assume an arbitrary MAC BC Dwn, and if the encoders E1,E2 are the encoders of a MAC-T code, then the maximum achievable rate is where S(Yk,Tk) denotes the (Shannon)T capacity of the k-th sub-point-to-point channel Yk with the constraint Tk. This theorem follows from (5) and (7) if all codes in the proof from Theorem 1 are restricted to transmission codes. The results above refer to a non-linear network coding system where both users send their coded messages to the relay communication device, which determines whether they are identical and sends back the result. This is non-linear because the (exactly one) result bit is not a linear combination of the received sequences. A linear relay can, for example, broadcast an XOR operation on its received signals. If this doesn't happen at the physical layer, rate-limited identity channels without noise are assumed. In the case of network coding at the physical layer, the noisy received signals are combined. Since the resulting sequences have the same length as the codewords, the broadcast phase is non-trivial. An XOR relay, or adder relay, which rounds up the signal Y1+Y2, behaves like an adder channel as described above, and since Yk = xk + Zk for k = 1, 2, the first communication device can calculate the difference, where the noise of the downlink channel to Alice 102 is given. Similarly, Bob 106 can calculate it. The task is CT over MACs k = 1, 2, i.e., ID over single-user channels. Thus, the capacity under the condition T1 × T2 is given by C(s, Wk, Tk × P(ℕ)) (for ID) or CT(s, Wk, Tk × P(ℕ)) (transmit) for each scale s. The T-capacitance for s = idℕ is Some examples are presented below. Example 1 is a communication device. The communication device may include: a processor configured to perform a procedure comprising: receiving a first consensus check code from a first communication device; receiving a second consensus check code from a second communication device; generating a combination code by network encoding the first consensus check code and the second consensus check code; and instructing the transmission of the combination code to the first communication device and to the second communication device; receiving a third consensus check code from a third communication device; and generating the combination code by network encoding the first consensus check code, the second consensus check code, and the third consensus check code. In Example 2, the subject of Example 1 may optionally exhibit that the network coding is linear. In Example 3, the subject of Example 1 may optionally exhibit that the network coding is non-linear. In Example 4, the object of Example 3 can optionally have a combination code with a length of exactly one bit. In Example 5, the item can optionally exhibit any of Examples 1 to 4 that the communication device is set up as a relay communication device. Example 6 is a communication device. The communication device may include: a processor configured to perform a procedure comprising: generating a first consensus check code from a first communication device; instructing the transmission of the first consensus check code to another communication device; receiving a combination code formed by network encoding the first consensus check code and a second consensus check code generated by yet another communication device; and checking the combination code, wherein the combination code was generated by network encoding the first consensus check code, the second consensus check code, and a third consensus check code generated by a third communication device. In Example 7, the subject of Example 6 may optionally exhibit that the network coding is linear. In Example 8, the subject of Example 6 may optionally exhibit that the network coding is non-linear. In Example 9, the object of Example 8 can optionally have a combination code with a length of exactly one bit. Example 10 is a system. The system may include: a communication device according to any of Examples 6 to 9; the first communication device set up according to any of Examples 6 to 9; and the second communication device set up according to any of Examples 6 to 9. Example 11 is a procedure. The procedure may include: receiving a first consensus check code from a first communication device; receiving a second consensus check code from a second communication device; generating a combination code by network encoding the first consensus check code and the second consensus check code; and instructing the transmission of the combination code to the first communication device and to the second communication device; receiving a third consensus check code from a third communication device; and generating the combination code by network encoding the first consensus check code, the second consensus check code, and the third consensus check code. In Example 12, the subject of Example 11 may optionally exhibit that the network encoding is linear. In Example 13, the subject of Example 11 may optionally exhibit that the network coding is non-linear. In Example 14, the object of Example 13 may optionally have a combination code with a length of exactly one bit. In Example 15, the object of any of Examples 11 to 14 may optionally include the fact that the procedure is carried out by a communication device. In Example 16, the subject of Example 15 may optionally include the procedure being carried out by a relay communication device. Example 17 is a procedure. The procedure may include: generating a first consensus check code from a first communication device; instructing the transmission of the first consensus check code to another communication device; receiving a combination code formed by network encoding the first consensus check code and a second consensus check code generated by yet another communication device; and checking the combination code, wherein the combination code was generated by network encoding the first consensus check code, the second consensus check code, and a third consensus check code generated by a third communication device. In Example 18, the subject of Example 17 may optionally exhibit that the network encoding is linear. In Example 19, the subject of Example 17 may optionally exhibit that the network coding is non-linear. In Example 20, the object of Example 19 may optionally have a combination code with a length of exactly one bit.

Claims

Communication device comprising: a processor configured to perform a procedure comprising: • receiving a first consensus check code from a first communication device; • receiving a second consensus check code from a second communication device; • generating a combination code by network encoding the first consensus check code and the second consensus check code; and • instructing the transmission of the combination code to the first communication device and to the second communication device; • receiving a third consensus check code from a third communication device; • generating the combination code by network encoding the first consensus check code, the second consensus check code, and the third consensus check code. Communication device according to claim 1, wherein the network encoding comprises linear network encoding. Communication device according to claim 1, wherein the network encoding comprises non-linear network encoding. Communication device according to claim 3, wherein the combination code has a length of exactly one bit. Communication device according to one of claims 1 to 4, configured as a relay communication device. Communication device comprising: a processor configured to perform a procedure comprising: • generating a first consensus check code from a first communication device; • instructing the transmission of the first consensus check code to another communication device; • receiving a combination code formed by network encoding the first consensus check code and a second consensus check code generated by yet another communication device; and • checking the combination code; • wherein the combination code was generated by network encoding the first consensus check code, the second consensus check code, and a third consensus check code generated by a third communication device. Communication device according to claim 6, wherein the network encoding comprises linear network encoding. Communication device according to claim 6, wherein the network encoding comprises non-linear network encoding. Communication device according to claim 8, wherein the combination code has a length of exactly one bit. A method comprising: • Receiving a first consensus verification code from a first communication device; • Receiving a second consensus verification code from a second communication device; • Generating a combination code by network encoding the first consensus verification code and the second consensus verification code; and • Instructing the transmission of the combination code to the first communication device and to the second communication device; • Receiving a third consensus verification code from a third communication device; • Generating the combination code by network encoding the first consensus verification code, the second consensus verification code, and the third consensus verification code. A method comprising: • generating a first consensus check code from a first communication device; • instructing the transmission of the first consensus check code to another communication device; • receiving a combination code formed by network encoding the first consensus check code and a second consensus check code generated by yet another communication device; and • checking the combination code; • wherein the combination code was generated by network encoding the first consensus check code, the second consensus check code, and a third consensus check code generated by a third communication device.

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