Griesmer type bounds for additive codes over finite fields, integral and fractional MDS codes

TL;DR

This paper establishes new Griesmer type bounds for additive MDS codes over finite fields, providing upper limits on code length and classifying fractional additive MDS codes, including a complete classification for size 243 over GF(9).

Contribution

It introduces novel bounds for additive MDS codes and classifies fractional additive MDS codes, expanding understanding of their length and structure.

Findings

New Griesmer type bounds for additive codes

Complete classification of fractional additive MDS codes of size 243 over GF(9)

Fractional MDS codes can surpass lengths of known integral MDS codes

Abstract

In this article we prove Griesmer type bounds for additive codes over finite fields. These new bounds give upper bounds on the length of maximum distance separable (MDS) codes, codes which attain the Singleton bound. We will also consider codes to be MDS if they attain the fractional Singleton bound, due to Huffman. We prove that this bound in the fractional case can be obtained by codes whose length surpasses the length of the longest known codes in the integral case. For small parameters, we provide exhaustive computational results for additive MDS codes, by classifying the corresponding (fractional) subspace-arcs. This includes a complete classification of fractional additive MDS codes of size 243 over the field of order 9.