# Convergence Rate of Empirical Spectral Distribution of Random Matrices from Linear Codes

## TL;DR

This paper proves that the empirical spectral distribution of certain random matrices from linear codes converges to the Marchenko-Pastur law at a rate of at least n^{-1/4}, expanding understanding of spectral convergence in coding theory.

## Contribution

It establishes a quantitative convergence rate for the spectral distribution of random matrices from linear codes, under conditions on the dual code's Hamming distance.

## Key findings

- Convergence rate of at least n^{-1/4} in probability
- Spectral distribution converges to Marchenko-Pastur law
- Applicable to linear codes with dual Hamming distance ≥ 5

## Abstract

It is known that the empirical spectral distribution of random matrices obtained from linear codes of increasing length converges to the well-known Marchenko-Pastur law, if the Hamming distance of the dual codes is at least 5. In this paper, we prove that the convergence in probability is at least of the order $n^{-1/4}$ where $n$ is the length of the code.