A characterization of low-weight words that span generalized Reed-Muller codes

We consider the generalized Reed-Muller code R_Fq (ρ, m) of order ρ and length q^m, m > 1, over the field F_q, where q = p^t for prime p and t ≥ 1. In particular, we are interested in the case that t > 1 (so that q is not prime), and the order ρ is at least q. As shown by Ding and Key, under these conditions, unless ρ is very large (i.e., ρ > (m - 1) (q-1) + p^t-1 - 2), the code is not spanned by its minimum-weight words. Furthermore, there was no known characterization of words with small weight that span the code. In this correspondence, we characterize a set of words that span the code, and show that their weight is upper-bounded by q^{⌈m(q-1)-ρ /q-q/p⌉}, which is at most quadratic in the weight of the minimum-weight words.