On the covering radius of first order generalized Reed-Muller codes

Let q = p, p a prime number. Let B m = Fq[X1, . . . , Xm]/(X q 1 −X1, . . . , X m −Xm); B m actually consists of all the functions from Fq to Fq. We identify B q m with F q q through the application B m → F m q P 7→ (P x∈Fmq For all b ∈ Fq , we denote by 1b the function in B m such that 1b(b) = 1 and for all x 6= b, 1b(x) = 0. The weight |P | of P ∈ B m is Card({x, P (x) 6= 0}). The Hamming distance in B m is denoted by d(., .). For 0 ≤ r ≤ m(q − 1), the rth order generalized Reed-Muller code of length q is Rq(r,m) = {P ∈ B m, deg(P ) ≤ r} where deg(P ) is the degree of the representant of P with degree at most q − 1 in each variable (see [8]). For all 0 ≤ r ≤ m(q − 1), the affine group GAm(Fq) acts on Rq(r,m) by its natural action and we have