Random Matrices From Linear Codes and Wigner’s Semicircle Law

Abstract

In this paper, we consider a new normalization of matrices obtained by choosing distinct codewords at random from linear codes over finite fields and find that under some natural algebraic conditions of the codes their empirical spectral distribution converges to Wigner’s semicircle law as the length of the codes goes to infinity. One such condition is that the dual distance of the codes is at least five. This is analogous to previous work on the empirical spectral distribution of similar matrices obtained in this fashion that converges to the Marchenko–Pastur law.