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^┌<span>m(q-1)-ρ/q-q/p┐/, which is at most quadratic in the weight of the minimum-weight words.