Griesmer type bounds for additive codes over finite fields, integral and fractional MDS codes
Simeon Ball, Michel Lavrauw, Tabriz Popatia
TL;DR
This work extends Griesmer-type bounds to additive codes over finite fields, deriving two novel bounds that constrain length given dimension and minimum distance, including fractional MDS cases. It uses a geometric framework based on projective systems and subspace arcs to translate code properties into hyperplane incidence conditions, enabling bounds and analysis of faithfulness. The authors prove bound-optimal conditions, explore duality, and present constructions showing attainability of bounds, along with comprehensive classifications of additive MDS codes over small fields. The results reveal additive codes can surpass linear MDS limits in length and provide a rich landscape of fractional MDS codes, with concrete classifications for $q\le 9$ and small parameters. These findings expand the design space for additive MDS codes and have potential implications for quantum coding and related applications.
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.