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This article is cited in 1 scientific paper (total in 1 paper)
On full-rank perfect codes over finite fields
A. M. Romanov Sobolev Institute of Mathematics, 4 Acad. Koptyug Ave., 630090 Novosibirsk, Russia
Abstract:
We propose a construction of full-rank $q$-ary $1$-perfect codes over finite fields. This is a generalization of the construction of full-rank binary $1$-perfect codes by Etzion and Vardy (1994). The properties of the $i$-components of q-ary Hamming codes are investigated and the construction of full-rank $q$-ary $1$-perfect codes is based on these properties. The switching construction of $1$-perfect codes is generalized for the $q$-ary case. We propose a generalization of the notion of $i$-component of a $1$-perfect code and introduce the concept of an $(i,\sigma)$-component of $q$-ary $1$-perfect codes. We also present a generalization of the Lindström–Schönheim construction of $q$-ary $1$-perfect codes and provide a lower bound for the number of pairwise distinct $q$-ary $1$-perfect codes of length $n$. Bibliogr. 16.
Keywords:
Hamming code, nonlinear perfect code, full-rank code, $i$-component.
DOI:
https://doi.org/10.17377/daio.2016.23.522
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English version:
Journal of Applied and Industrial Mathematics, 2016, 10:3, 444–452
Bibliographic databases:
UDC:
519.8 Received: 29.12.2015 Revised: 17.03.2016
Citation:
A. M. Romanov, “On full-rank perfect codes over finite fields”, Diskretn. Anal. Issled. Oper., 23:3 (2016), 107–123; J. Appl. Industr. Math., 10:3 (2016), 444–452
Citation in format AMSBIB
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\jour J. Appl. Industr. Math.
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\pages 444--452
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Linking options:
http://mi.mathnet.ru/eng/da855 http://mi.mathnet.ru/eng/da/v23/i3/p107
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This publication is cited in the following articles:
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A. M. Romanov, “On non-full-rank perfect codes over finite fields”, Designs Codes Cryptogr., 87:5 (2019), 995–1003
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