Let $F$ be a countable infinite field. Consider the space $F^{{\mathbb N}_0}$ of all sequences $u=(u_1,u_2,\dots)$, where $u_i\in F$ and $u_i=0$ except a finite set of indices $i\in\mathbb N$. A perfect $F$-valued code $C\subset F^{{\mathbb N}_0}$ of infinite length with Hamming distance $3$ can be defined in a standard way. For each $m\in\mathbb N$ ($m\geqslant 2$), we define a Hamming code $H_F^{(m)}$ using a checking matrix with $m$ rows. Also, we define one more Hamming code $H_F^{(\omega)}$ using a checking matrix with countable rows. Then we prove (Theorem 1) that all these Hamming codes are nonequivalent. In spite of this fact, Theorem 2 asserts that any perfect linear code $C\subset F^{{\mathbb N}_0}$ is affinely equivalent to one of the Hamming codes $H_F^{(m)}$, $m=2,3,\dots,\omega$. For the code $H_F^{(\omega)}$, we construct a continuum of nonequivalent checking matrices having countable rows (Theorem 4). Also, for this code, a countable family of nonequivalent checking matrices with columns having finite supports is constructed. Further, Theorem 8 asserts that a checking matrix with countable rows and columns with finite supports has a minimal checking submatrix.
Keywords:
perfect $F$-valued code, code of infinite length, checking matrix, complete system of triples.
S. A. Malyugin. Linear perfect codes of infinite length over infinite fields. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 1165-1182. http://geodesic.mathdoc.fr/item/SEMR_2020_17_a72/
@article{SEMR_2020_17_a72,
author = {S. A. Malyugin},
title = {Linear perfect codes of~infinite length over~infinite fields},
journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
pages = {1165--1182},
year = {2020},
volume = {17},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SEMR_2020_17_a72/}
}
TY - JOUR
AU - S. A. Malyugin
TI - Linear perfect codes of infinite length over infinite fields
JO - Sibirskie èlektronnye matematičeskie izvestiâ
PY - 2020
SP - 1165
EP - 1182
VL - 17
UR - http://geodesic.mathdoc.fr/item/SEMR_2020_17_a72/
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%0 Journal Article
%A S. A. Malyugin
%T Linear perfect codes of infinite length over infinite fields
%J Sibirskie èlektronnye matematičeskie izvestiâ
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%U http://geodesic.mathdoc.fr/item/SEMR_2020_17_a72/
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