On the structure of some arcs related to caps and the nonexistence of some optimal codes

Abstract

In this paper we solve two instances of the main problem in coding theory for linear codes of dimension 5 over ${\mathbb{F}}_4$. We prove the nonexistence of $[395,5,295{]}_4$- and $[396,5,296{]}_4$-codes which implies the exact values $n_4(5,295)=396$ and $n_4(5,296)=397$. As a by-product, we characterize the arcs with parameters $(100,26)$ in $PG(3,4)$.