The density of weights of Generalized Reed--Muller codes

TL;DR

This paper investigates the distribution of weights in Generalized Reed--Muller codes, revealing that the set of possible relative weights is sparse and characterized by specific rational forms, especially as the number of variables grows.

Contribution

It provides a novel analysis of the density of codeword weights in Reed--Muller codes, showing the sparsity and characterizing all possible distributions of weights.

Findings

The set of relative weights is sparse, with gaps around irrationals not of the form l/p^k.

For any rational of the form l/p^k, weights can be approximated.

Complete characterization of distributions from constant degree polynomials.

Abstract

We study the density of the weights of Generalized Reed--Muller codes. Let $RM_p(r,m)$ denote the code of multivariate polynomials over $\F_p$ in $m$ variables of total degree at most $r$. We consider the case of fixed degree $r$, when we let the number of variables $m$ tend to infinity. We prove that the set of relative weights of codewords is quite sparse: for every $\alpha \in [0,1]$ which is not rational of the form $\frac{\ell}{p^k}$, there exists an interval around $\alpha$ in which no relative weight exists, for any value of $m$. This line of research is to the best of our knowledge new, and complements the traditional lines of research, which focus on the weight distribution and the divisibility properties of the weights. Equivalently, we study distributions taking values in a finite field, which can be approximated by distributions coming from constant degree polynomials, where we do not bound the number of variables. We give a complete characterization of all such distributions.