Some Algebraic Questions about the Reed-Muller Code

Abstract

Let $R_q(r,n)$ denote the $r$th order Reed-Muller code of length $q^n$ over ${\mathbb{F}}_q$. We consider two algebraic questions about the Reed-Muller code. Let $H_q(r,n)=R_q(r,n)/R_q(r-1,n)$. (1) When $q=2$, it is known that there is a "duality" between the actions of $\text{GL}(n,{\mathbb{F}}_2)$ on $H_2(r,n)$ and on $H_2(r^{\prime },n)$, where $r+r^{\prime }=n$. The result is false for a general $q$. However, we find that a slightly modified duality statement still holds when $q$ is a prime or $r<\text{char}{\mathbb{F}}_q$. (2) Let $\mathcal{F}({\mathbb{F}}_q^n,{\mathbb{F}}_q)$ denote the ${\mathbb{F}}_q$-algebra of all functions from ${\mathbb{F}}_q^n$ to ${\mathbb{F}}_q$. It is known that when $q$ is a prime, the Reed-Muller codes $\{0\}=R_q(-1,n)\subset R_q(0,n)\subset \cdots \subset R_q(n(q-1),n)=\mathcal{F}({\mathbb{F}}_q^n,{\mathbb{F}}_q)$ are the only $\text{AGL}(n,{\mathbb{F}}_q)$-submodules of $\mathcal{F}({\mathbb{F}}_q^n,{\mathbb{F}}_q)$. In particular, $H_q(r,n)$ is an irreducible $\text{GL}(n,{\mathbb{F}}_q)$-module when $q$ is a prime. For a general $q$, $H_q(r,n)$ is not necessarily irreducible. We determine all its submodules and the factors in its composition series. The factors of the composition series of $H_q(r,n)$ provide an explicit family of irreducible representations of $\text{GL}(n,{\mathbb{F}}_q)$ over ${\mathbb{F}}_q$.

21 pages, 2 figures, 2 tables