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Fundamentalnaya i Prikladnaya Matematika, 2013, Volume 18, Issue 4, Pages 137–154
(Mi fpm1535)
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This article is cited in 4 scientific papers (total in 4 papers)
Basic Reed–Muller codes as group codes
I. N. Tumaykin Lomonosov Moscow State University, Moscow, Russia
Abstract:
Reed–Muller codes are one of the most well-studied families of codes, however, there are still open problems regarding their structure. Recently, a new ring-theoretic approach has emerged that provides a rather intuitive construction of these codes. This approach is centered around the notion of basic Reed–Muller codes. We recall that Reed–Muller codes over a prime field are radical powers of a corresponding group algebra. In this paper, we prove that basic Reed–Muller codes in the case of a nonprime field of arbitrary characteristic are distinct from radical powers. This implies the same result for regular codes. Also we show how to describe the inclusion graph of basic Reed–Muller codes and radical powers via simple arithmetic equations.
Citation:
I. N. Tumaykin, “Basic Reed–Muller codes as group codes”, Fundam. Prikl. Mat., 18:4 (2013), 137–154; J. Math. Sci., 206:6 (2015), 699–710
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https://www.mathnet.ru/eng/fpm1535 https://www.mathnet.ru/eng/fpm/v18/i4/p137
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Abstract page: | 287 | Full-text PDF : | 132 | References: | 37 |
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