FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2013, VOLUME 18, NUMBER 4, PAGES 137-154

I. N. Tumaykin

Abstract

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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.

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