Pseudorandom Bits for Polynomials
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
Worst Case to Average Case Reductions for Polynomials
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
| Hi-index | 754.84 |
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 α which is not rational of the form α = l/qk, there exists some ε 0 such that no weights fall in the interval (α - ε, α + ε). This demonstrates a new property of the weight distribution of Generalized Reed-Muller codes.