Additive codes attaining the Griesmer bound

TL;DR

This paper proves that additive codes can always meet a Griesmer-type bound for large minimum distance, leading to many optimal codes that outperform linear codes.

Contribution

It establishes a general condition under which additive codes attain the Griesmer bound, advancing the understanding of their optimal parameters.

Findings

Additive codes can meet the Griesmer bound with equality for large minimum distance.

Many infinite series of additive codes outperform linear codes.

The paper provides explicit constructions for such optimal additive codes.

Abstract

Additive codes may have better parameters than linear codes. However, still very few cases are known and the explicit construction of such codes is a challenging problem. Here we show that a Griesmer type bound for the length of additive codes can always be attained with equality if the minimum distance is sufficiently large. This solves the problem for the optimal parameters of additive codes when the minimum distance is large and yields many infinite series of additive codes that outperform linear codes.