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 F-q in m variables of total degree at most r. The relative weight of a codeword f is an element of RMq(r, m) is the fraction of nonzero entries in f. The possible relative weights are studied, when the field F-q 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 a which is not rational of the form alpha = l/q(k), there exists some epsilon > 0 such that no weights fall in the interval (alpha - epsilon, alpha + epsilon). This demonstrates a new property of the weight distribution of Generalized Reed-Muller codes.