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IEEE Information Theory Workshop (ITW)

Local Correctabilities and Dual Codes of Symmetric Reed–Muller Codes

Wei Yan, Sian-Jheng Lin

Published Oct 17, 2021• 0 citations
Publisher

Abstract

Recently, a variant of the Reed-Muller family, called symmetric Reed-Muller codes, was proposed. In this paper, we prove that multivariable symmetric Reed-Muller codes form a class of locally-correctable codes. Then, the dual codes of symmetric Reed-Muller codes are given.
DOI
10.1109/itw48936.2021.9611491

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Reed-Muller (RM) codes, a classical family of codes known for their elegant algebraic structure, have recently been shown to achieve capacity under maximum-likelihood (ML) decoding on the binary erasure channel and this has rekindled interest in their efficient decoding. We consider the code family RM(m-3,m) and develop a new ML decoder, for transmission over the binary symmetric channel, that exploits their large symmetry group. The new decoder has lower complexity than an earlier method introduced by Seroussi and Lempel in 1983.
In this thesis, we start from the theory of classical error-correcting codes. We also introduce classical Reed-Muller codes along with its properties. We discuss the encoding and decoding methods for classical Reed-Muller codes using the majority rule. Then we turn to the topic of quantum error-correcting codes.We talk about stabilizer codes and discuss how classical Reed-Muller codes can be transformed into quantum Reed-Muller codes using Steane's method. Finally, we give a construction of a quantum code with parameter [[8,3,3]], and discuss its encoding and decoding process.
In this paper, by treating Reed-Muller (RM) codes as a special class of low-density parity-check (LDPC) codes and assuming that sub-blocks of the parity-check matrix are randomly interleaved to each other as Gallager's codes, we present a short proof that RM codes are entropy-achieving as source coding for Bernoulli sources and capacity-achieving as channel coding for binary memoryless symmetric (BMS) channels, also known as memoryless binary-input output-symmetric (BIOS) channels, in terms of bit error rate (BER) under maximum-likelihood (ML) decoding.