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English version:
Journal of Applied and Industrial Mathematics, 2016, 10:3, 444-452

Volume 23, No 3, 2016, P. 107-123

UDC 519.8
A. M. Romanov
On full-rank perfect codes over finite fields

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: 10.17377/daio.2016.23.522

Alexander M. Romanov 1
1. Sobolev Institute of Mathematics,
4 Koptyug Ave., 630090 Novosibirsk, Russia
e-mail: rom@math.nsc.ru

Revised 17 March 2016

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