In practice, since many communication networks are huge in scale, or complicated in structure, or even dynamic, the predesigned linear network codes based on the network topology is impossible even if the topological structure is known. Therefore, random linear network coding has been proposed as an acceptable coding technique for the case that the network topology cannot be utilized completely. Motivated by the fact that different network topological information can be obtained for different practical applications, we study the performance analysis of random linear network coding by analyzing some failure probabilities depending on these different topological information of networks. We obtain some tight or asymptotically tight upper bounds on these failure probabilities and indicate the worst cases for these bounds, i.e., the networks meeting the upper bounds with equality. In addition, if the more topological information of the network is utilized, the better upper bounds are obtained. On the other hand, we also discuss the lower bounds on the failure probabilities.
We consider the variable-to-fixed length lossy source coding (VFSC) problem. The optimal compression rate of the average length of variable-to-fixed source coding, allowing a non-vanishing probability of excess-distortion $\varepsilon$, is shown to be equal to $(1-\varepsilon)R(D)$, where $R(D)$ is the rate-distortion function of the source. In comparison to the related results of Koga and Yamamoto as well as Kostina, Polyanskiy, and Verdú for fixed-to-variable length source coding, our results demonstrate an interesting feature that variable-to-fixed length source coding has the same first-or…
Sparse Network Coding (SNC) has been a promising network coding scheme as an improvement for Random Linear Network Coding (RLNC) in terms of the computational complexity. However, in this literature, there has been no analytical expressions for the probability of decoding a fraction of source messages after transmission of some coded packets. In this work, we looked into the problem of the probability of decoding a fraction of source messages, i.e., partial decoding, in the decoder for a system which uses SNC. We exploited the Principle of Inclusion and Exclusion to derive expressions of parti…
The variable-length source coding problem allowing the error probability up to some constant is considered for general sources. In this problem the optimum mean codeword length of variable-length codes has already been determined. On the other hand, in this paper, we focus on the overflow (or excess codeword length) probability instead of the mean codeword length. The infimum of overflow thresholds under the constraint that both of the error probability and the overflow probability are smaller than or equal to some constant is called the optimum overflow threshold. In this paper, we first deri…
This paper considers a network where a node wishes to transmit a source message to a legitimate receiver in the presence of an eavesdropper. The transmitter secures its transmissions employing a sparse implementation of Random Linear Network Coding (RLNC). A tight approximation to the probability of the eavesdropper recovering the source message is provided. The proposed approximation applies to both the cases where transmissions occur without feedback or where the reliability of the feedback channel is impaired by an eavesdropper jamming the feedback channel. An optimization framework for min…
The exact order of the optimal sub-exponentially decaying factor in the classical bounds on the error probability of fixed-length codes over a Gallager-symmetric discrete memoryless channel with and without ideal feedback is determined. Regardless of the availability of feedback, it is shown that the order of the optimal sub-exponential factor exhibits a dichotomy. Moreover, the proof technique is used to establish the third-order term in the normal approximation for symmetric channels, where a similar dichotomy is shown to exist.
Sparse random linear network coding (SRLNC) used as a class of erasure codes to ensure the reliability of multicast communications has been widely investigated. However, an exact expression for the decoding success probability of SRLNC is still unknown, and existing expressions are either asymptotic or approximate. In this paper, we derive an exact expression for the decoding success probability of SRLNC. The key to achieving this is to propose a criterion that a vector is contained in a subspace. To obtain this criterion, we construct a basis of a subspace, with respect to this basis, the coo…
Reliable and low latency multicast communication is important for future vehicular communication. Sparse random linear network coding approach used to ensure the reliability of multicast communication has been widely investigated. A fundamental problem of such communication is to characterize the decoding success probability, which is given by the probability of a sparse random matrix over a finite field being full rank. However, the exact expression for the probability of a sparse random matrix being full rank is still unknown, and existing approximations are recursive or not consistently tig…
Many data compressors regularly encode probability distributions for entropy coding - requiring minimal description length type of optimizations. Canonical prefix/Huffman coding usually just writes lengths of bit sequences, this way approximating probabilities with powers-of-2. Operating on more accurate probabilities usually allows for better compression ratios, and is possible e.g. using arithmetic coding and Asymmetric Numeral Systems family. Especially the multiplication-free tabled variant of the latter (tANS) builds automaton often replacing Huffman coding due to better compression at si…
We consider the one helper source coding problem posed and investigated by Ahlswede, Körner and Wyner. In this system, the error probability of decoding goes to one as the source block length $n$ goes to infinity. This implies that we have a strong converse theorem for the one helper source coding problem. In this paper we provide a much stronger version of this strong converse theorem for the one helper source coding problem. We prove that the error probability of decoding tends to one exponentially and derive an explicit lower bound of this exponent function.
This paper considers the achievability and converse bounds on the maximal channel coding rate at a given blocklength and error probability over AWGN channels. The problem stems from covert communication with Gaussian codewords. By re-visiting [18], we first present new and more general achievability bounds for random coding schemes under maximal or average probability of error requirements. Such general bounds are then applied to covert communication in AWGN channels where codewords are generated from Gaussian distribution while meeting the maximal power constraint. Further comparison is made …
The context tree source is a source model in which the occurrence probability of symbols is determined from a finite past sequence, and is a broader class of sources that includes i.i.d. and Markov sources. The proposed source model in this paper represents that a subsequence in each interval is generated from a different context tree model. The Bayes code for such sources requires weighting of the posterior probability distributions for the change patterns of the context tree source and for all possible context tree models. Therefore, the challenge is how to reduce this exponential order comp…
Revisiting Shannon's Source Coding Theorem with Distributional Uncertainty under the Nonlinear Expectation Theory
National Center for Mathematics and Interdisciplinary Sciences, Academy of Mathematics and Systems Science, CAS（中国科学院数学与系统科学研究院交叉科学中心）；School of Information and Communication Engineering, Beijing University of Posts and Telecommunications（北京邮电大学信息与通信工程学院）；University of Chinese Academy of Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Science（中国科学院大学数学与系统科学研究院）
