Holes in Generalized Reed–Muller Codes

The possible relative weights of codewords of Generalized Reed-Muller codes are studied. Let RMq(r,m) denote the code of polynomials over the finite field Fq in m variables of total degree at most r. The relative weight of a codeword f ¿ RMq(r,m) is the fraction of nonzero entries in f. The possible relative weights are studied, when the field Fq and the degree r are fixed, and the number of variables m tends to infinity. It is proved that the set of possible weights is sparse-for any ¿ which is not rational of the form ¿ = ¿/q ^k, there exists some ¿ > 0 such that no weights fall in the interval (¿-¿,¿+¿). This demonstrates a new property of the weight distribution of Generalized Reed-Muller codes.