Optimal binary and ternary linear codes with hull dimension one

Abstract

Hulls of linear codes have been of interest and extensively studied due to their wide applications. In this paper, we focus on constructions and optimality of linear codes with hull dimension one over small finite fields. General constructions for such codes are given together with the analysis on their parameters. Optimal linear $$[n,2,d]_q$$ codes with hull dimension one are presented for all positive integers $$n\ge 3$$ and $$q\in \{2,3\}$$ . Moreover, for $$q=2$$ , the enumeration of such optimal codes is given up to equivalence.