Some algebraic questions about the Reed-Muller code.

Let R q (r , n) denote the r th order Reed-Muller code of length q n over 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 GL (n , 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 < char F q. (2) Let F (F q n , F q) denote the F q -algebra of all functions from F q n to F q. It is known that when q is a prime, the Reed-Muller codes { 0 } = R q (− 1 , n) ⊂ R q (0 , n) ⊂ ⋯ ⊂ R q (n (q − 1) , n) = F (F q n , F q) are the only AGL (n , F q) -submodules of F (F q n , F q). In particular, H q (r , n) is an irreducible GL (n , 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 GL (n , F q) over F q. [ABSTRACT FROM AUTHOR]