There exists no self-dual [24,12,10] code over $${{\mathbb F}_5}$$

Abstract

Self-dual codes over $${{\mathbb F}_5}$$ 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 $${{\mathbb F}_5}$$ , using the classification of 24-dimensional odd unimodular lattices due to Borcherds.