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.
@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/
LA - en
ID - SEMR_2020_17_a72
ER -
%0 Journal Article
%A S. A. Malyugin
%T Linear perfect codes of infinite length over infinite fields
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2020
%P 1165-1182
%V 17
%U http://geodesic.mathdoc.fr/item/SEMR_2020_17_a72/
%G en
%F SEMR_2020_17_a72
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/
