Some algebraic questions about the Reed-Muller code

Abstract

Let Rq(r,n) denote the rth order Reed-Muller code of length qn over Fq. We consider two algebraic questions about the Reed-Muller code. Let Hq(r,n)=Rq(r,n)/Rq(r−1,n). (1) When q=2, it is known that there is a “duality” between the actions of GL(n,F2) on H2(r,n) and on H2(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<charFq. (2) Let F(Fqn,Fq) denote the Fq-algebra of all functions from Fqn to Fq. It is known that when q is a prime, the Reed-Muller codes {0}=Rq(−1,n)⊂Rq(0,n)⊂⋯⊂Rq(n(q−1),n)=F(Fqn,Fq) are the only AGL(n,Fq)-submodules of F(Fqn,Fq). In particular, Hq(r,n) is an irreducible GL(n,Fq)-module when q is a prime. For a general q, Hq(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 Hq(r,n) provide an explicit family of irreducible representations of GL(n,Fq) over Fq.