We provide a general framework for bounding the block error threshold of a linear code$C \subseteq \mathbb{F}_{2}^{N}$over the erasure channel in terms of its bit error threshold. Our approach relies on understanding the minimum support weight of any$r$-dimensional subcode of$C$, for all small values of$r$. As a proof of concept, we use our machinery to obtain a new proof of the celebrated result that Reed-Muller codes achieve capacity on the erasure channel with respect to block error probability.
We provide a general framework for bounding the block error threshold of a linear codeC⊆F2Nover the erasure channel in terms of its bit error threshold. Our approach relies on understanding the minimum support weight of anyr-dimensional subcode ofC, for all small values ofr. As a proof of concept, we use our machinery to obtain a new proof of the celebrated result that Reed-Muller codes achieve capacity on the erasure channel with respect to block error probability.
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.
A method for estimating the block error probability of Reed-Muller codes on a binary erasure channel and a binary symmetric channel under a permutation decoding algorithm is proposed. This method is based on an analysis of the block error probability under a recursive decoding algorithm for a fixed number of input erasures or errors. Simulation results demonstrate that the proposed method allows to effectively predict the block error rate performance of Reed-Muller codes under permutation decoding with a different number of permutations used.
Reed-Muller codes encode an m-variate polynomial of degree r by evaluating it on all points in {0,1}m. We denote this code by RM(m,r). The minimal distance of RM(m,r) is 2m−r and so it cannot correct more than half that number of errors in the worst case. For random errors one may hope for a better result. In this work we give an efficient algorithm (in the block length n=2m) for decoding random errors in Reed-Muller codes far beyond the minimal distance. Specifically, for low rate codes (of degree r=o(m)) we can correct a random set of (1/2−o(1))n errors with high probability. For high rate codes (of degree m−r for r=o(m/logm)), we can correct roughly mr/2 errors. More generally, for any integer r, our algorithm can correct any error pattern in RM(m,m−(2r+2)) for which the same erasure pattern can be corrected in RM(m,m−(r+1)). The results above are obtained by applying recent results of Abbe, Shpilka and Wigderson (STOC, 2015), Kumar and Pfister (2015) and Kudekar et al. (2015) regarding the ability of Reed-Muller codes to correct random erasures. The algorithm is based on solving a carefully defined set of linear equations and thus it is significantly different than other algorithms for decoding Reed-Muller codes that are based on the recursive structure of the code. It can be seen as a more explicit proof of a result of Abbe et al. that shows a reduction from correcting erasures to correcting errors, and it also bares some similarities with the famous Berlekamp-Welch algorithm for decoding Reed-Solomon 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.
We present an approach to showing that a linear code is resilient to random errors. We use this approach to obtain decoding results for both transitive codes and Reed-Muller codes. We give three kinds of results about linear codes in general, and transitive linear codes in particular. 1) We give a tight bound on the weight distribution of every transitive linear code C⊆F2N: Prc∈C[∣c∣=αN]≤2−(1−h(α))dim(C). 2) We give a criterion that certifies that a linear code C can be decoded on the binary symmetric channel. Let Ks(x) denote the Krawtchouk polynomial of degree s, and let C⊥ denote the dual code of C. We show that bounds on Ec∈C⊥[KεN(∣c∣)2] imply that C recovers from errors on the binary symmetric channel with parameter ε. Weaker bounds can be used to obtain list-decoding results using similar methods. One consequence of our criterion is that whenever the weight distribution of C⊥ is sufficiently close to the binomial distribution in some interval around 2N, C is resilient to ε-errors. 3) We combine known estimates for the Krawtchouk polynomials with our weight bound for transitive codes, and with known weight bounds for Reed-Muller codes, to obtain list-decoding results for both these families of codes. In some regimes, our bounds for Reed-Muller codes achieve the information-theoretic optimal trade-off between rate and list size.
The question of whether Reed-Muller (RM) codes achieve capacity on binary memoryless symmetric (BMS) channels has drawn attention since it was resolved positively for the binary erasure channel by Kudekar et al. in 2016. In 2021, Reeves and Pfister extended this to prove the bit-error probability vanishes on BMS channels when the code rate is less than capacity. In 2023, Abbe and Sandon improved this to show the block-error probability also goes to zero. These results analyze decoding functions using symmetry and the nested structure of RM codes. In this work, we focus on binary-input symmetric classicalquantum (BSCQ) channels and the Holevo capacity. For a BSCQ, we consider observables that estimate the channel input in the sense of minimizing the mean-squared error (MSE). Using the orthogonal decomposition of these observables under a weighted inner product, we establish a recursive relation for the minimum MSE estimate of a single bit in the RM code. Our results show that any set of2o(logN)bits can be decoded with high probability when the code rate is less than the Holevo capacity.