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Diskretn. Anal. Issled. Oper., 2016, Volume 23, Number 3, Pages 107–123 (Mi da855)  

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.

Funding Agency Grant Number
Russian Foundation for Basic Research 14-01-00507


DOI: https://doi.org/10.17377/daio.2016.23.522

Full text: PDF file (299 kB)
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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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\by A.~M.~Romanov
\paper On full-rank perfect codes over finite fields
\jour Diskretn. Anal. Issled. Oper.
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\vol 23
\issue 3
\pages 107--123
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\crossref{https://doi.org/10.17377/daio.2016.23.522}
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\transl
\jour J. Appl. Industr. Math.
\yr 2016
\vol 10
\issue 3
\pages 444--452
\crossref{https://doi.org/10.1134/S1990478916030157}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84983527195}


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