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This article is cited in 2 scientific papers (total in 2 papers)
Coding Theory
On intersections of Reed–Muller like codes
F. I. Solov'eva Sobolev Institute of Mathematics, Siberian Branch
of the Russian Academy of Sciences, Novosibirsk, Russia
Abstract:
A binary code that has the parameters and possesses the main properties of the classical $r$ th-order Reed–Muller code $RM_{r,m}$ will be called an $r$ th-order Reed–Muller like code and will be denoted by $LRM_{r,m}$. The class of such codes contains the family of codes obtained by the Pulatov construction and also classical linear and $\mathbb{Z}_4$-linear Reed–Muller codes. We analyze the intersection problem for the Reed–Muller like codes. We prove that for any even $k$ in the interval $0\le k\le 2^{2\sum\limits_{i=0}^{r-1}\binom{m-1}{i}}$ there exist $LRM_{r,m}$ codes of order $r$ and length $2^m$ having intersection size $k$. We also prove that there exist two Reed–Muller like codes of order $r$ and length $2^m$ whose intersection size is $2k_1 k_2$ with $1\le k_s\le |RM_{r-1,m-1}|$, $s\in\{1,2\}$, for any admissible length starting from $16$.
Keywords:
Reed–Muller code, Reed–Muller like code, code intersection problem, Pulatov codes, components of Reed–Muller codes, $i$-component, switching, switching construction for codes.
Received: 25.06.2021 Revised: 10.11.2021 Accepted: 10.11.2021
Citation:
F. I. Solov'eva, “On intersections of Reed–Muller like codes”, Probl. Peredachi Inf., 57:4 (2021), 63–73; Problems Inform. Transmission, 57:4 (2021), 357–367
Linking options:
https://www.mathnet.ru/eng/ppi2355 https://www.mathnet.ru/eng/ppi/v57/i4/p63
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