Holes in Generalized Reed–Muller Codes

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