# Random Matrices from Linear Codes and Wigner's Semicircle Law II

## TL;DR

This paper proves that random matrices derived from linear codes over finite fields converge to Wigner's semicircle law, with a convergence rate depending on the code length, under the condition that the dual distance is at least 5.

## Contribution

It establishes that a dual distance of at least 5 guarantees spectral convergence to Wigner's law with a specific convergence rate, extending previous results.

## Key findings

- Spectral distribution converges to Wigner's semicircle law as code length increases.
- Convergence rate is of order $n^{-eta}$ for some $0<eta<1$.
- Dual distance ≥ 5 is sufficient for convergence.

## Abstract

Recently we considered a class of random matrices obtained by choosing distinct codewords at random from linear codes over finite fields and proved that under some natural algebraic conditions their empirical spectral distribution converges to Wigner's semicircle law as the length of the codes goes to infinity. One of the conditions is that the dual distance of the codes is at least 5. In this paper, employing more advanced techniques related to Stieltjes transform, we show that the dual distance being at least 5 is sufficient to ensure the convergence, and the convergence rate is of the form $n^{-\beta}$ for some $0 < \beta < 1$, where $n$ is the length of the code.