Эта публикация цитируется в
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Дискретная математика и математическая кибернетика
Linear perfect codes of infinite length over infinite fields
S. A. Malyugin Sobolev Institute of Mathematics, 4, Koptyuga ave., Novosibirsk, 630090, Russia
Аннотация:
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.
Ключевые слова:
perfect $F$-valued code, code of infinite length, checking matrix, complete system of triples.
УДК:
519.72
MSC: 94B60 Поступила 11 декабря 2019 г., опубликована
24 августа 2020 г.
Язык публикации: английский
DOI:
10.33048/semi.2020.17.088