The ML redundancy of a code is defined as the smallest number of rows in its parity-check matrix such that a message-passing decoder working in the corresponding Tanner graph achieves maximum-likelihood (ML) performance on an erasure channel. General upper bounds on ML redundancy are obtained. In particular, it is shown that the ML redundancy of a q-ary code is at most the number of minimal codewords in its dual code, divided by q−1. Special upper bounds are derived for codes whose dual code contains a covering design. For example, the ML redundancy of a Simplex code of length n is shown to be no greater than (n2− 4n + 9)/6.
The ML redundancy of a code is defined as the smallest number of rows in its parity-check matrix such that a message-passing decoder working in the corresponding Tanner graph achieves maximum-likelihood (ML) performance on an erasure channel. General upper bounds on ML redundancy are obtained. In particular, it is shown that the ML redundancy of a q-ary code is at most the number of minimal codewords in its dual code, divided by q−1. Special upper bounds are derived for codes whose dual code contains a covering design. For example, the ML redundancy of a Simplex code of length n is shown to be no greater than (n2− 4n + 9)/6.
Stopping sets play a crucial role in failure events of iterative decoders over a binary erasure channel (BEC). The ℓth stopping redundancy is the minimum number of rows in the parity-check matrix of a code, which contains no stopping sets of size up to ℓ. In this paper, a notion of coverable stopping sets is defined. In order to achieve maximum-likelihood performance under iterative decoding over the BEC, the parity-check matrix should contain no coverable stopping sets of size ℓ, for 1 ≤ ℓ ≤ n-k, where n is the code length, k is the code dimension. By estimating the number of coverable stopping sets, we obtain upper bounds on the ℓth stopping redundancy, 1 ≤ ℓ ≤ n-k. The bounds are derived for both specific codes and code ensembles. In the range 1 ≤ ℓ ≤ d-1, for specific codes, the new bounds improve on the results in the literature. Numerical calculations are also presented.
It is now well known that the performance of a linear code C under iterative decoding on a binary erasure channel (and other channels) is determined by the size of the smallest stopping set in the Tanner graph for C. Several recent papers refer to this parameter as the \emph{stopping distance} s of C. This is somewhat of a misnomer since the size of the smallest stopping set in the Tanner graph for C depends on the corresponding choice of a parity-check matrix. It is easy to see that s≤d, where d is the minimum Hamming distance of C, and we show that it is always possible to choose a parity-check matrix for C (with sufficiently many dependent rows) such that s=d. We thus introduce a new parameter, termed the \emph{stopping redundancy} of C, defined as the minimum number of rows in a parity-check matrix H for C such that the corresponding stopping distance s(H) attains its largest possible value, namely s(H)=d. We then derive general bounds on the stopping redundancy of linear codes. We also examine several simple ways of constructing codes from other codes, and study the effect of these constructions on the stopping redundancy. Specifically, for the family of binary Reed-Muller codes (of all orders), we prove that their stopping redundancy is at most a constant times their conventional redundancy. We show that the stopping redundancies of the binary and ternary extended Golay codes are at most 35 and 22, respectively. Finally, we provide upper and lower bounds on the stopping redundancy of MDS codes.
It is now well known that the performance of a linear code Copf under iterative decoding on a binary erasure channel (and other channels) is determined by the size of the smallest stopping set in the Tanner graph for Copf. Several recent papers refer to this parameter as the stopping distance s of Copf. This is somewhat of a misnomer since the size of the smallest stopping set in the Tanner graph for Copf depends on the corresponding choice of a parity-check matrix. It is easy to see that s les d, where d is the minimum Hamming distance of Copf, and we show that it is always possible to choose a parity-check matrix for Copf (with sufficiently many dependent rows) such that s=d. We thus introduce a new parameter, the stopping redundancy of Copf, defined as the minimum number of rows in a parity- check matrix H for Copf such that the corresponding stopping distance s(H) attains its largest possible value, namely, s(H)=d. We then derive general bounds on the stopping redundancy of linear codes. We also examine several simple ways of constructing codes from other codes, and study the effect of these constructions on the stopping redundancy. Specifically, for the family of binary Reed-Muller codes (of all orders), we prove that their stopping redundancy is at most a constant times their conventional redundancy. We show that the stopping redundancies of the binary and ternary extended Golay codes are at most 34 and 22, respectively. Finally, we provide upper and lower bounds on the stopping redundancy of MDS codes
It is now well known that the performance of a linear code Copf under iterative decoding on a binary erasure channel (and other channels) is determined by the size of the smallest stopping set in the Tanner graph for Copf. Several recent papers refer to this parameter as the stopping distance s of Copf. This is somewhat of a misnomer since the size of the smallest stopping set in the Tanner graph for Copf depends on the corresponding choice of a parity-check matrix. It is easy to see that s les d, where d is the minimum Hamming distance of Copf, and we show that it is always possible to choose a parity-check matrix for Copf (with sufficiently many dependent rows) such that s = d. We thus introduce a new parameter, termed the stopping redundancy of Copf, defined as the minimum number of rows in a parity-check matrix H for Copf such that the corresponding stopping distance s(H) attains its largest possible value, namely s(H) = d. We then derive general bounds on the stopping redundancy of linear codes. We also examine several simple ways of constructing codes from other codes, and study the effect of these constructions on the stopping redundancy. Specifically, for the family of binary Reed-Muller codes (of all orders), we prove that their stopping redundancy is at most a constant times their conventional redundancy. We show that the stopping redundancies of the binary and ternary extended Golay codes are at most 34 and 22, respectively. Finally, we provide upper and lower bounds on the stopping redundancy of MDS codes
Key to the success of modern error correcting codes is the effectiveness of message-passing iterative decoding (MPID). Unlike maximum-likelihood (ML) decoding, the performance of MPID depends not only on the code, but on how the code is represented. In particular, the performance of MPID is potentially improved by using a redundant representation. We focus on Tanner graphs and study combinatorial structures therein that help explain the performance disparity among different representations of the same code. Emphasis is placed on the complexity- performance tradeoff, as more and more check nodes are allowed in the graph. Our discussion applies to MPID as well as linear programming decoding (LPD), which we collectively refer to as graph-based decoding. On an erasure channel, it is well-known that the performance of MPID or LPD is determined by stopping sets. Following Schwartz and Vardy, we define the stopping redundancy as the smallest number of check nodes in a Tanner graph such that smallest size of a non-empty stopping set is equal to the minimum Hamming distance of the code. Roughly speaking, stopping redundancy measures the complexity requirement (in number of check nodes) for MPID of a redundant graph representation to achieve performance comparable to ML decoding (up to a constant factor for small channel erasure probability). General upper bounds on stopping redundancy are obtained. One of our main contribution is a new upper bound based on probabilistic analysis, which is shown to be by far the strongest. From this bound, it can be shown, for example, that for a fixed minimum distance, the stopping redundancy grows just linearly with the redundancy (codimension). Specific results on the stopping redundancy of Golay and Reed-Muller codes are also obtained. We show that the stopping redundancy of maximum distance separable (MDS) codes is bounded in between a Turan number and a single-exclusion (SE) number --- a purely combinatorial quantity that we introduce. By studying upper bounds on the SE number, new results on the stopping redundancy of MDS codes are obtained. Schwartz and Vardy conjecture that the stopping redundancy of an MDS code should only depend on its length and minimum distance. Our results provide partial confirmation, both exact and asymptotic, to this conjecture. Stopping redundancy can be large for some codes. We observe that significantly fewer checks are needed if a small number of small stopping sets are allowed. These small stopping sets can then be dealt with by ̀g̀uessing'' during the iterative decoding process. Correspondingly, the guess-g stopping redundancy is defined and it is shown that the savings in number of required check nodes are potentially significant. Another theoretically interesting question is when MPID of a Tanner graph achieves the same word error rate an ML decoder. This prompts us to define and study ML redundancy. Applicability and possible extensions of the current work to a non-erasure channel are discussed. A framework based on pseudo-codewords is considered and shown to be relevant. However, it is also observed that the polytope characterization of pseudo-codewords is not complete enough to be an accurate indicator of MPID performance. Finally, in a separate piece of work, the probability of undetected error (PUE) for over-extended Reed-Solomon codes is studied through the weight distribution bounds of the code. The resulting PUE expressions are shown to be tight in a well-defined sense
Linear block codes over noisy channels causing both erasures and errors can be decoded by deleting the erased symbols and decoding the resulting vector with respect to a punctured code and then retrieving the erased symbols. This can be accomplished using separating parity-check matrices. For a given maximum number of correctable erasures, such matrices yield parity-check equations that do not check any of the erased symbols and which are sufficient to characterize all punctured codes corresponding to this maximum number of erasures. Separating parity-check matrices typically have redundant rows. An upper bound on the minimum number of rows in separating parity-check matrices, which is called the separating redundancy, is derived which proves that the separating redundancy tends to behave linearly as a function of the code length.