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arXiv:2201.11617 · IEEE International Symposium on Information Theory (ISIT)

List Decoding of 2-Interleaved Binary Alternant Codes

Chih‐Chiang Huang, Hedongliang Liu, Lukas Holzbaur, Sven Puchinger, Antonia Wachter-Zeh

Published Jun 26, 2022• 1 citation
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Abstract

This paper is concerned with list decoding of 2-interleaved binary alternant codes. The principle of the proposed algorithm is based on a combination of a list decoding algorithm for (interleaved) Reed-Solomon codes and an algorithm for (non-interleaved) alternant codes. A new upper bound on the decoding radius is derived and the list size is shown to scale polynomially in the code parameters. While it remains an open problem whether this upper bound is achievable, the provided simulation results show that a decoding radius exceeding the binary Johnson radius can be achieved with a high probability of decoding success by the proposed algorithm.

DOI
10.1109/isit50566.2022.9834689

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This paper is concerned with list decoding of 2-interleaved binary alternant codes. The principle of the proposed algorithm is based on a combination of a list decoding algorithm for (interleaved) Reed-Solomon codes and an algorithm for (non-interleaved) alternant codes. A new upper bound on the decoding radius is derived and the list size is shown to scale polynomially in the code parameters. While it remains an open problem whether this upper bound is achievable, the provided simulation results show that a decoding radius exceeding the binary Johnson radius can be achieved with a high probability of decoding success by the proposed algorithm.
Let a binary Reed-Muller code RM(s;m) of length n be used on a memoryless channel with an input alphabet ±1 and a real-valued output ℝ. Given a received vector y in ℝn; we define its generalized distance T to any codeword c as the sum Σ|yj|taken over all positions j, in which vectors y, c have opposite signs. We then consider the list ℒTof codewords located within distance T from the received vector y and estimate the size LTof this list using the generalized Johnson bound. For any RM code RM(s,m) of fixed order s, the algorithm is proposed that performs list decoding beyond the error-correcting radius with linear complexity in length n and retrieves the code list ℒTwith complexity of order nsLTfor any decoding radius T within the generalized Johnson bound.
List decoding of block codes is an alternative approach to thedecoding problem with appealing qualities. The fairly recent development of efficient algorithms for list decoding of Reed-Solomon codes spur new fuel to the study of this decoding strategy. In this paper we give a weight-based characterization of the set of correctable error patterns under list-of-2 ecodingof (τ,2(\tau, 2(τ,2)-list-decodable linear codes with known weight distribution. We apply our characterization of the set of correctable error patterns to a few codes in a family of low-rate list-of-2 decodable Reed-Solomon codes. We study the increase in error-correction performance obtained in a symmetric AWGN channel by using list-of-2 decoding instead of traditional decoding for these codes. Some simulation results for list-of-2 decoding on QAM channels using the Guruswami-Sudan algorithm for decoding of Reed-Solomon codes are alsopresented.
We present a decoding algorithm for concatenated codes when the outer code is a Reed-Solomon code and the inner code is arbitrary. "Soft" information on the reliability of various symbols is passed by the inner decodings and exploited in the Reed-Solomon decoding. This is the first analysis of such a soft algorithm that works for arbitrary inner codes; prior analyses could only, handle some special inner codes. Crucial to our analysis is a combinatorial result on the coset weight distribution of codes given only its minimum distance. Our result enables us to decode essentially up to the "Johnson radius" of a concatenated code when the outer distance is large (the Johnson radius is the "a priori list decoding radius" of a code as a function of its distance). As a consequence, we are able to present simple and efficient constructions of q-ary linear codes that are list decodable up to a fraction (1 - 1/q - /spl epsiv/) of errors and have rate /spl Omega/(/spl epsiv//sup 6/). Codes that can correct such a large fraction of errors have found numerous complexity-theoretic applications. The previous constructions of linear codes with a similar rate used algebraic-geometric codes and thus suffered from a complicated construction and slow decoding.
Non-binary linear block codes (NB-LBCs) are an important class of error-correcting codes that are especially competent in correcting burst errors. They have broad applications in modern communications and storage systems. However, efficient soft-decision decoding of these codes remains to be further developed. This paper proposes successive cancellation list (SCL) decoding for NB-LBCs that are defined over a finite field of characteristic two, i.e., F2r, whereris the extension degree. By establishing a one-to-r mapping between the binary composition of each non-binary codeword andrbinary polar codewords, SCL decoding of therpolar codes can be performed with a complexity that is sub-quadratic in the codeword length. A simplified path sorting is further proposed to facilitate the decoding. Simulation results on short-length extended Reed-Solomon (eRS) and non-binary extended BCH (NB-eBCH) codes show that SCL decoding can outperform their state-of-the-art soft-decision decoding with fewer finite field arithmetic operations. For length-16 eRS codes, their maximum-likelihood (ML) decoding performances can be approached with a moderate list size.
Given a classical error-correcting block code, the task of quantum list decoding is to produce from any quantumly corrupted codeword a short list containing all messages whose codewords exhibit high "presence" in the quantumly corrupted codeword.Efficient quantum list decoders have been used to prove a quantum hardcore property of classical codes.However, the code rates of all known families of efficiently quantum list-decodable codes are, unfortunately, too small for other practical applications.To improve those known code rates, we prove that a specific code family of polynomially small code rate over a fixed code alphabet, obtained by concatenating generalized Reed-Solomon codes as outer codes with Hadamard codes as inner codes, has an efficient quantum list-decoding algorithm if its codewords have relatively high codeword presence in a given quantumly corrupted codeword.As an immediate application, we use the quantum list decodability of this code family to solve a certain form of quantum search problems in polynomial time.When the codeword presence becomes smaller, in contrast, we show that the quantum list decodability of generalized Reed-Solomon codes with high confidence is closely related to the efficient solvability of the following two problems: the noisy polynomial interpolation problem and the bounded distance vector problem.Moreover, assuming that NP BQP, we also prove that no efficient quantum list decoder exists for the generalized Reed-Solomon codes.
Reed-Muller codes are a popular code family based on the Plotkin construction. Recently, these codes have regained some interest due to their close relation to polar codes and their low-complexity decoding. We consider a similar code family, i.e., the Plotkin concatenation with binary BCH component codes. This construction is more flexible regarding the attainable code parameters. In this work, we consider a list-based decoding algorithm for the Plotkin concatenation with BCH component codes. The proposed list decoding leads to a significant coding gain with only a small increase in computational complexity. Simulation results demonstrate that the Plotkin concatenation with the proposed decoding achieves near maximum likelihood decoding performance. This coding scheme can outperform polar codes for moderate code lengths.