Recently, sequences of error-correcting codes with doubly-transitive permutation groups were shown to achieve capacity on erasure channels under symbol-wise maximum a posteriori (MAP) decoding. From this, it follows that Reed-Muller and primitive narrow-sense BCH codes achieve capacity in the same setting. In this article, we extend this result to a large family of cyclic codes by considering codes whose permutation groups satisfy a condition weaker than double transitivity. The article combines two simple technical contributions. First, we show that the transition width of a monotone boolean function is O(1/log k), where k is the size of the smallest orbit induced by its symmetry group. The proof is based on Talagrand's lower bound on influences for monotone boolean functions. Second, we consider the extrinsic information transfer (EXIT) function of an Fq-linear cyclic code whose blocklength N divides qt- 1 and is coprime with q - 1. We show that this EXIT function is a monotone boolean function whose symmetry group contains no orbits of size smaller than the smallest prime divisor of t. Combining these, we show that sequences of cyclic codes, whose blocklengths satisfy the above conditions, achieve capacity on the q-ary erasure channel if all prime divisors of t tend to infinity.
Recently, sequences of error-correcting codes with doubly-transitive permutation groups were shown to achieve capacity on erasure channels under symbol-wise maximum a posteriori (MAP) decoding. From this, it follows that Reed-Muller and primitive narrow-sense BCH codes achieve capacity in the same setting. In this article, we extend this result to a large family of cyclic codes by considering codes whose permutation groups satisfy a condition weaker than double transitivity. The article combines two simple technical contributions. First, we show that the transition width of a monotone boolean function is O(1/log k), where k is the size of the smallest orbit induced by its symmetry group. The proof is based on Talagrand's lower bound on influences for monotone boolean functions. Second, we consider the extrinsic information transfer (EXIT) function of an Fq-linear cyclic code whose blocklength N divides qt- 1 and is coprime with q - 1. We show that this EXIT function is a monotone boolean function whose symmetry group contains no orbits of size smaller than the smallest prime divisor of t. Combining these, we show that sequences of cyclic codes, whose blocklengths satisfy the above conditions, achieve capacity on the q-ary erasure channel if all prime divisors of t tend to infinity.
We introduce a new approach to proving that a sequence of deterministic linear codes achieves capacity on an erasure channel under maximum a posteriori decoding. Rather than relying on the precise structure of the codes our method exploits code symmetry. In particular, the technique applies to any sequence of linear codes where the blocklengths are strictly increasing, the code rates converge, and the permutation group of each code is doubly transitive. In other words, we show that symmetry alone implies near-optimal performance. An important consequence of this result is that a sequence of Reed-Muller codes with increasing blocklength and converging rate achieves capacity. This possibility has been suggested previously in the literature but it has only been proven for cases where the limiting code rate is 0 or 1. Moreover, these results extend naturally to all affine-invariant codes and, thus, to extended primitive narrow-sense BCH codes. This also resolves, in the affirmative, the existence question for capacity-achieving sequences of binary cyclic codes. The primary tools used in the proof are the sharp threshold property for symmetric monotone boolean functions and the area theorem for extrinsic information transfer functions.
The question whether RM codes are capacity-achieving is a long-standing open problem in coding theory that was recently answered in the affirmative for transmission over erasure channels [1], [2]. Remarkably, the proof does not rely on specific properties of RM codes, apart from their symmetry. Indeed, the main technical result consists in showing that any sequence of linear codes, with doubly-transitive permutation groups, achieves capacity on the memoryless erasure channel under bit-MAP decoding. Thus, a natural question is what happens under block-MAP decoding. In [1], [2], by exploiting further symmetries of the code, the bit-MAP threshold was shown to be sharp enough so that the block erasure probability also converges to 0. However, this technique relies heavily on the fact that the transmission is over an erasure channel. We present an alternative approach to strengthen results regarding the bit-MAP threshold to block-MAP thresholds. This approach is based on a careful analysis of the weight distribution of RM codes. In particular, the flavor of the main result is the following: assume that the bit-MAP error probability decays as N−δ, for some δ > 0. Then, the block-MAP error probability also converges to 0. This technique applies to transmission over any binary memoryless symmetric channel. Thus, it can be thought of as a first step in extending the proof that RM codes are capacity-achieving to the general case.
The past decade has seen notable advances in our understanding of structured error-correcting codes, particularly binary Reed-Muller (RM) codes. While initial breakthroughs were for erasure channels based on symmetry, extending these results to the binary symmetric channel (BSC) and other binary memoryless symmetric (BMS) channels required new tools and conditions. Recent work uses nesting to obtain multiple weakly correlated looks at each code bit to establish capacity-achieving performance under bit-MAP and block-MAP decoding. This paper revisits and extends past approaches, aiming to simplify proofs, unify insights, remove unnecessary conditions, and provide new results. By leveraging powerful results from the analysis of boolean functions, we derive recursive bounds using two or three looks at each stage. This gives bounds on the bit-error probability that decay exponentially in the number of stages. For the BSC, we incorporate level-k inequalities and hypercontractive techniques to achieve the faster decay rate required for vanishing block-error probability. The same ideas also extend to product codes with RM component codes, which are transitive but not doubly-transitive in general, and yield vanishing bit-error and block-error probability at rates arbitrarily close to capacity. The results are presented in a semi-tutorial style, providing both theoretical insights and practical implications for future research on structured codes.
Consider a binary linear code of length N, minimum distance d-{\min}, transmission over the binary erasure channel with parameter 0 < \epsilon < 1 or the binary symmetric channel with parameter 0 < \epsilon < \frac{1}{2}, and block-MAP decoding. It was shown by Tillich and Zemor that in this case the error probability of the block-MAP decoder transitions 'quickly' from \delta to 1-\delta for any \delta > 0 if the minimum distance is large. In particular the width of the transition is of order O(1/\sqrt{d-{\min}}). We strengthen this result by showing that under suitable conditions on the weight distribution of the code, the transition width can be as small as \Theta(1/N^{\frac{1}{2}-\kappa}), for any \kappa > 0, even if the minimum distance of the code is not linear. This condition applies e.g., to Reed-Mueller codes. Since \Theta(1/N^{\frac{1}{2}}) is the smallest transition possible for any code, we speak of 'almost' optimal scaling. We emphasize that the width of the transition says nothing about the location of the transition. Therefore this result has no bearing on whether a code is capacity-achieving or not. As a second contribution, we present a new estimate on the derivative of the EXIT function, the proof of which is based on the Blowing-Up Lemma.
Abstract — We introduce the notion of the stopping redundancy hierarchy of a linear block code as a measure of the trade-off between performance and complexity of iterative decoding for the binary erasure channel. We derive lower and upper bounds for the stopping redundancy hierarchy via Lovász’s Local Lemma and Bonferroni-type inequalities, and specialize them for codes with cyclic parity-check matrices. Based on the observed prop-erties of parity-check matrices with good stopping redundancy characteristics, we develop a novel decoding technique, termed automorphism group decoding, that combines iterative message passing and permutation decoding. We also present bounds on the smallest number of permutations of an automorphism group decoder needed to correct any set of erasures up to a prescribed size. Simulation results demonstrate that for a large number of algebraic codes, the performance of the new decoding method is close to that of maximum likelihood decoding.
We introduce the notion of the stopping redundancy hierarchy of a linear block code as a measure of the tradeoff between performance and complexity of iterative decoding for the binary erasure channel. We derive lower and upper bounds for the stopping redundancy hierarchy via Lovasz's local lemma (LLL) and Bonferroni-type inequalities, and specialize them for codes with cyclic parity-check matrices. Based on the observed properties of parity-check matrices with good stopping redundancy characteristics, we develop a novel decoding technique, termed automorphism group decoding, that combines iterative message passing and permutation decoding. We also present bounds on the smallest number of permutations of an automorphism group decoder needed to correct any set of erasures up to a prescribed size. Simulation results demonstrate that for a large number of algebraic codes, the performance of the new decoding method is close to that of maximum-likelihood (ML) decoding.