Optimum Linear Codes with Support Constraints over Small Fields

TL;DR

This paper demonstrates that optimal linear codes with support constraints can be constructed from Reed-Solomon codes over small fields, confirming the GM-MDS conjecture and advancing coding theory.

Contribution

It proves that the largest minimum distance under support constraints is achievable via subcodes of Reed-Solomon codes, settling the GM-MDS conjecture.

Findings

Optimal codes can be derived from Reed-Solomon codes over small fields.

The GM-MDS conjecture is confirmed.

Supports the design of efficient error-correcting codes with constraints.

Abstract

We consider the problem of designing optimal linear codes (in terms of having the largest minimum distance) subject to a support constraint on the generator matrix. We show that the largest minimum distance can be achieved by a subcode of a Reed-Solomon code of small field size. As a by-product of this result, we settle the GM-MDS conjecture of Dau et. al. in the affirmative.