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 $\Bbb F_q$. 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 $\text{GL}(n,\Bbb F_2)$ on $H_2(r,n)$ and on $H_2(r',n)$, where $r+r'=n$. 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 $r<\text{char}\,\Bbb F_q$. (2) Let $\mathcal F(\Bbb F_q^n,\Bbb F_q)$ denote the $\Bbb F_q$-algebra of all functions from $\Bbb F_q^n$ to $\Bbb F_q$. It is known that when $q$ is a prime, the Reed-Muller codes ${0}=R_q(-1,n)\subset R_q(0,n)\subset\cdots\subset R_q(n(q-1),n)=\mathcal F(\Bbb F_q^n,\Bbb F_q)$ are the only $\text{AGL}(n,\Bbb F_q)$-submodules of $\mathcal F(\Bbb F_q^n,\Bbb F_q)$. In particular, $H_q(r,n)$ is an irreducible $\text{GL}(n,\Bbb F_q)$-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 $\text{GL}(n,\Bbb F_q)$ over $\Bbb F_q$.