On the Weights of General MDS Codes

TL;DR

This paper investigates the weight spectra of MDS codes over arbitrary alphabets, establishing which weights are realized and proving that all binary MDS codes are equivalent to linear ones.

Contribution

It characterizes the weight spectra of all q-ary MDS codes with certain parameters and proves the equivalence of binary MDS codes to linear codes.

Findings

All k weights from n to n-k+1 are realized for q-ary MDS codes containing zero.

The case n=q+k-1 is fully determined.

All binary MDS codes are equivalent to linear MDS codes.

Abstract

The weight spectra of MDS codes of length $ n $ and dimension $ k $ over the arbitrary alphabets are studied. For all $ q $-ary MDS codes of dimension $ k $ containing the zero codeword, it is shown that all $ k $ weights from $ n $ to $ n-k+1 $ are realized. The remaining case $ n=q+k-1 $ is also determined. Additionally, we prove that all binary MDS codes are equivalent to linear MDS codes. The proofs are combinatorial, and self contained.