Random Matrices from Linear Codes and Wigner's semicircle law

TL;DR

This paper demonstrates that matrices constructed from random codewords of linear codes over finite fields exhibit spectral distributions converging to Wigner's semicircle law under certain algebraic conditions, extending previous spectral law results.

Contribution

It introduces a new normalization method for matrices from linear codes and proves convergence to Wigner's law under dual distance conditions, expanding spectral law understanding.

Findings

Spectral distribution converges to Wigner's semicircle law

Dual distance of at least 5 is sufficient

Extends previous results from Marchenko-Pastur law to Wigner's 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 5. 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.