There exists no self-dual [24,12,10] code over F5

TL;DR

This paper proves that a self-dual [24,12,10] code over F5 does not exist by leveraging the classification of 24-dimensional odd unimodular lattices, filling a gap in the understanding of such codes.

Contribution

It establishes the non-existence of a specific self-dual code over F5 at length 24, using lattice classification techniques.

Findings

No self-dual [24,12,10] code over F5 exists.

The largest minimum weight for self-dual codes at length 24 is at most 9.

The result completes the classification of self-dual codes over F5 for length 24.

Abstract

Self-dual codes over F5 exist for all even lengths. The smallest length for which the largest minimum weight among self-dual codes has not been determined is 24, and the largest minimum weight is either 9 or 10. In this note, we show that there exists no self-dual [24,12,10] code over F5, using the classification of 24-dimensional odd unimodular lattices due to Borcherds.