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.

Bibliographie

[1] S.V Avgustinovich, F.I. Solov'eva, “On the nonsystematic perfect binary codes”, Problems Inform. Transmission, 32:3 (1996), 258–261 | MR | Zbl

[2] A.N. Kolmogorov, S.V. Fomin, Elements of function theory and functional analysis, Nauka, M., 1976 (in Russian) | MR

[3] S.A. Malyugin, “Perfect binary codes of infinite length”, J. Appl. Indust. Math., 8:4 (2017), 552–556 | DOI | MR

[4] S.A. Malyugin, “Perfect binary codes of infinite length with complete system of triples”, Sib. Elektron. Mat. Izv., 14 (2017), 877–888 | MR | Zbl

[5] S.A. Malyugin, “Systemanic and nonsystematic perfect codes of infinite length over finite fields”, Sib. Elektron. Mat. Izv., 16 (2019), 1732–1751 | DOI | MR | Zbl

[6] K.T. Phelps, M.J. LeVan, “Nonsystematic perfect codes”, SIAM J. Discrete Math., 12:1 (1999), 27–34 | DOI | MR | Zbl