Geodesic
Linear perfect codes of~infinite length over~infinite fields
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 1165-1182.

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@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},
     publisher = {mathdoc},
     volume = {17},
     year = {2020},
     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
PB  - mathdoc
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
%I mathdoc
%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/

[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