Characterization and classification of optimal ternary linear one-dimensional hull codes

It has been shown that the effectiveness of some algorithms involving permutation code equivalence depends on hulls of linear codes. This paper focuses on studying ternary linear one-dimensional hull codes, and further considers the largest minimum distance \(d_1(n, k)\) $d_1(n,k)$ among all ternary linear one-dimensional hull [n, k] codes for \(n\le 20\) $n\le 20$ or \(k \le 3\) $k\le 3$ . Most importantly, we classify optimal ternary linear one-dimensional hull [n, 2] codes. Further, we construct some ternary linear one-dimensional hull codes with better minimum distances compared with known results.