Approximate MDS Property of Linear Codes

TL;DR

This paper investigates the weight spectrum of linear codes with large field sizes, demonstrating that most such codes have spectra closely resembling MDS codes, using probabilistic methods.

Contribution

It introduces a probabilistic approach to show that nearly all linear codes with super-linear field sizes have weight spectra similar to MDS codes, revealing their approximate MDS property.

Findings

Most linear codes with super-linear field size have spectra close to MDS codes

Probabilistic methods effectively demonstrate the approximate MDS property

The weight spectrum of these codes is nearly optimal

Abstract

In this paper, we study the weight spectrum of linear codes with \emph{super-linear} field size and use the probabilistic method to show that for nearly all such codes, the corresponding weight spectrum is very close to that of a maximum distance separable (MDS) code.