On Bases of BCH Codes with Designed Distance 3 and Their Extensions

I. Yu Mogilnykh; F. I. Solov’eva

doi:10.1134/S003294602004002X

On Bases of BCH Codes with Designed Distance 3 and Their Extensions

Research output: Contribution to journal › Article › peer-review

Abstract

We consider narrow-sense BCH codes of length p^m − 1 over F_p, m ≥ 3. We prove that neither such a code with designed distance δ = 3 nor its extension for p ≥ 5 is generated by the set of its codewords of the minimum nonzero weight. We establish that extended BCH codes with designed distance δ = 3 for p ≥ 3 are generated by the set of codewords of weight 5, where basis vectors can be chosen from affine orbits of some codewords.

Original language	English
Pages (from-to)	309-316
Number of pages	8
Journal	Problems of Information Transmission
Volume	56
Issue number	4
DOIs	https://doi.org/10.1134/S003294602004002X
Publication status	Published - Dec 2020

Keywords

affine-invariant code
BCH code
cyclic code
minimum weight basis
single orbit affine generator

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10.1134/S003294602004002X

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title = "On Bases of BCH Codes with Designed Distance 3 and Their Extensions",

abstract = "We consider narrow-sense BCH codes of length pm − 1 over Fp, m ≥ 3. We prove that neither such a code with designed distance δ = 3 nor its extension for p ≥ 5 is generated by the set of its codewords of the minimum nonzero weight. We establish that extended BCH codes with designed distance δ = 3 for p ≥ 3 are generated by the set of codewords of weight 5, where basis vectors can be chosen from affine orbits of some codewords.",

keywords = "affine-invariant code, BCH code, cyclic code, minimum weight basis, single orbit affine generator",

author = "Mogilnykh, {I. Yu} and Solov{\textquoteright}eva, {F. I.}",

note = "Funding Information: The research was supported in part by the Ministry of Science and Higher Education of the Russian Federation, contract no. 075-02-2020-1479/1. Publisher Copyright: {\textcopyright} 2020, Pleiades Publishing, Inc. Copyright: Copyright 2021 Elsevier B.V., All rights reserved.",

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AU - Mogilnykh, I. Yu

AU - Solov’eva, F. I.

N1 - Funding Information: The research was supported in part by the Ministry of Science and Higher Education of the Russian Federation, contract no. 075-02-2020-1479/1. Publisher Copyright: © 2020, Pleiades Publishing, Inc. Copyright: Copyright 2021 Elsevier B.V., All rights reserved.

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N2 - We consider narrow-sense BCH codes of length pm − 1 over Fp, m ≥ 3. We prove that neither such a code with designed distance δ = 3 nor its extension for p ≥ 5 is generated by the set of its codewords of the minimum nonzero weight. We establish that extended BCH codes with designed distance δ = 3 for p ≥ 3 are generated by the set of codewords of weight 5, where basis vectors can be chosen from affine orbits of some codewords.

AB - We consider narrow-sense BCH codes of length pm − 1 over Fp, m ≥ 3. We prove that neither such a code with designed distance δ = 3 nor its extension for p ≥ 5 is generated by the set of its codewords of the minimum nonzero weight. We establish that extended BCH codes with designed distance δ = 3 for p ≥ 3 are generated by the set of codewords of weight 5, where basis vectors can be chosen from affine orbits of some codewords.

KW - affine-invariant code

KW - BCH code

KW - cyclic code

KW - minimum weight basis

KW - single orbit affine generator

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