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Advances in Mathematics of Communications

A criterion for decoding on the binary symmetric channel

Anup Rao, Oscar Sprumont

Published Feb 28, 2024• 2 citations
Publisher

Abstract

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 and doubly transitive 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 \subseteq \mathbb{F}_2^N $:$ \mathop {\Pr }\limits_{c \in C}[ \text{wt}(c) = \alpha N] \leq 2^{-(1-h(\alpha)) \mathsf{dim}(C)}. $2. We give a criterion that certifies that a linear code $ C $ can be decoded on the binary symmetric channel. Let $ K_s(x) $ denote the Krawtchouk polynomial of degree $ s $, and let $ C^\perp $ denote the dual code of $ C $. We show that bounds on $ \mathbb{E}_{c \in C^{\perp}}[ K_{\epsilon N}( \text{wt}(c))^2] $ imply that $ C $ recovers from errors on the binary symmetric channel with parameter $ \epsilon $. 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^\perp $ is sufficiently close to the binomial distribution in some interval around $ \frac{N}{2} $, $ C $ is resilient to $ \epsilon $-errors.3. We combine known estimates for the Krawtchouk polynomials with our weight bound for transitive codes, and with known weight bounds for doubly transitive codes, to obtain list-decoding results for both these families of codes. In some regimes, our bounds for doubly transitive codes achieve the information-theoretic optimal trade-off between rate and list size.

DOI
10.3934/amc.2024007

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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 and doubly transitive 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⊆F2NC \subseteq \mathbb{F}_2^NC⊆F2N​:Pr⁡c∈C[wt(c)=αN]≤2−(1−h(α))dim(C).\mathop {\Pr }\limits_{c \in C}[ \text{wt}(c) = \alpha N] \leq 2^{-(1-h(\alpha)) \mathsf{dim}(C)}.c∈CPr​[wt(c)=αN]≤2−(1−h(α))dim(C).2. We give a criterion that certifies that a linear code CCC can be decoded on the binary symmetric channel. Let Ks(x)K_s(x)Ks​(x) denote the Krawtchouk polynomial of degree sss, and let C⊥C^\perpC⊥ denote the dual code of CCC. We show that bounds on Ec∈C⊥[KϵN(wt(c))2]\mathbb{E}_{c \in C^{\perp}}[ K_{\epsilon N}( \text{wt}(c))^2]Ec∈C⊥​[KϵN​(wt(c))2] imply that CCC recovers from errors on the binary symmetric channel with parameter ϵ\epsilonϵ. 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⊥C^\perpC⊥ is sufficiently close to the binomial distribution in some interval around N2\frac{N}{2}2N​, CCC is resilient to ϵ\epsilonϵ-errors.3. We combine known estimates for the Krawtchouk polynomials with our weight bound for transitive codes, and with known weight bounds for doubly transitive codes, to obtain list-decoding results for both these families of codes. In some regimes, our bounds for doubly transitive codes achieve the information-theoretic optimal trade-off between rate and list size.
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⊆F2NC \subseteq \mathbb{F}_2^NC⊆F2N​: Pr⁡c∈C[∣c∣=αN]≤2−(1−h(α))dim(C)\Pr_{c \in C}[|c| = αN] \leq 2^{-(1-h(α)) \mathsf{dim}(C)}Prc∈C​[∣c∣=αN]≤2−(1−h(α))dim(C). 2) We give a criterion that certifies that a linear code CCC can be decoded on the binary symmetric channel. Let Ks(x)K_s(x)Ks​(x) denote the Krawtchouk polynomial of degree sss, and let C⊥C^\perpC⊥ denote the dual code of CCC. We show that bounds on Ec∈C⊥[KεN(∣c∣)2]\mathbb{E}_{c \in C^{\perp}}[ K_{εN}(|c|)^2]Ec∈C⊥​[KεN​(∣c∣)2] imply that CCC recovers from errors on the binary symmetric channel with parameter εεε. 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⊥C^\perpC⊥ is sufficiently close to the binomial distribution in some interval around N2\frac{N}{2}2N​, CCC is resilient to εεε-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.
We study relationships between worst-case and random-noise properties of error correcting codes. More concretely, we consider connections between minimum distance, list decoding radius, and block error probability on noisy channels. A recent result of Pernice, Sprumont, and Wootters established the tight connection between list decoding radius and symmetric channel performance for linear codes. We extend this result to general codes. The proof proceeds by directly bounding the weight distribution rather than by sharp threshold techniques. We then turn to the relation between minimum distance and symmetric channel performance. The results we just described imply that a qqq-ary code of relative distance δδδ has vanishing error probability on the symmetric channel up to the Johnson radius Jq(δ)J_q(δ)Jq​(δ). We improve upon this bound in the case of linear codes, for a range δδδ, for q≥4q\ge 4q≥4. In our proof we consider the \emph{erasure} properties of codes, and bound their weight distribution through inequalities introduced by Samorodnitsky. The latter part of the proof gives a more general technique that bounds the symmetric channel performance of a linear code with constant relative distance and good erasure channel performance.
We strengthen the results from a recent work by the second author, achieving bounds on the weight distribution of binary linear codes that are successful under block-MAP (as well as bit-MAP) decoding on the BEC. We conclude that a linear code that is successful on the BEC can also decode over a range of binary memoryless symmetric (BMS) channels. In particular, applying the result of Kudekar, Kumar, Mondelli, Pfister, Şaşoğlu and Urbanke from STOC 2016, we prove that a Reed–Muller code of positive rate R decodes errors on the p with high probability if p < 1/2 − √2−R(1−2−R).
Bounds on the error probability of maximum likelihood decoding of a binary linear code are considered. The bounds derived use the weight spectrum of the code and they are tighter than the conventional union bound in the case of large noise in the channel. The bounds derived are applied to a code with an average spectrum, and the result is compared to the random coding exponent. The author shows that the bound considered for the binary symmetrical channel case coincides asymptotically with the random coding bound. For the case of AWGN channel the author shows that Berlekamp's (1980) tangential bound can be improved, but even this improved bound does not coincide with the random coding bound, although it can be very close to it.>
The performance of optimum linear block codes has been examined by computer simluation, and the error distribution at the decoder output, the block-error probability and the bit-error probability have been determined for the codes when they are used on a binary symmetric channel with random errors. A table showing the distribution of errors in the decoder output is presented, and it is shown how the table can be used to determine exact values for the bit-error probability and the block-error probability. The exact values of bit-error probability computed from the table are compared with values obtained using a known theoretical approximation to the bit-error probability, and some conclusions are drawn as to the general usefulness of the theoretical approximation. Also, block-error probabilities are computed from the information presented in the table and these values are compared with the estimates of block-error probability obtained from minimumdistance considerations.
The Poltyrev bound provides a very tight upper bound on the decoding error probability when using binary linear codes for transmission over the binary symmetric channel and the additive white Gaussian noise channel, making use of the code's weight spectrum. In the present work, the bound is extended to symmetric binary-input memoryless channels with a discrete output alphabet. The derived bound is demonstrated on a hybrid BSC-BEC channel. Additionally, a reduced-complexity bound is introduced at the cost of some loss in tightness.