Linear codes in the folded Hamming distance and the quasi MDS property

TL;DR

This paper investigates linear codes over subfields with folded Hamming distance, introduces quasi MDS codes, and provides bounds, constructions, and examples that outperform classical MDS codes in certain cases.

Contribution

It introduces the concept of quasi MDS codes in the folded Hamming setting and offers new bounds, constructions, and explicit examples surpassing traditional MDS codes.

Findings

Established bounds on code lengths relative to field sizes.

Provided explicit binary constructions with optimal lengths.

Demonstrated codes that outperform classical MDS codes.

Abstract

In this work, we study linear codes with the folded Hamming distance, or equivalently, codes with the classical Hamming distance that are linear over a subfield. This includes additive codes. We study MDS codes in this setting and define quasi MDS (QMDS) codes and dually QMDS codes, which attain a more relaxed variant of the classical Singleton bound. We provide several general results concerning these codes, including restriction, shortening, weight distributions, existence, density, geometric description and bounds on their lengths relative to their field sizes. We provide explicit examples and a binary construction with optimal lengths relative to their field sizes, which beats any MDS code.