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.
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The author is grateful to I.Yu. Mogilnykh for fruitful discussions and to a reviewer for a number of valuable remarks, which helped to improve the presentation.
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The research was carried out at the Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences under State Assignment no. 0314-2019-0016.
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Translated from Problemy Peredachi Informatsii, 2021, Vol. 57, No. 4, pp. 63–73 https://doi.org/10.31857/S0555292321040057.
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Solov’eva, F. On Intersections of Reed–Muller Like Codes. Probl Inf Transm 57, 357–367 (2021). https://doi.org/10.1134/S0032946021040050
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DOI: https://doi.org/10.1134/S0032946021040050