Linear perfect codes of~infinite length over~infinite fields

S. A. Malyugin

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 1165-1182

Résumé

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.

Détail

Keywords: perfect $F$-valued code, code of infinite length, checking matrix, complete system of triples.