Double Circulant Self-Dual Codes over \(GF(5)\)

All distinct double circulant self-dual codes over \(\text{GF}(5)\), with a minimum weight which is highest among all double circulant self-dual codes, have been found for each length \(n \leq 24\). For lengths \(14\), \(16\), and \(20\), these codes are extremal. In this paper, we characterize these extremal double circulant self-dual codes. In particular, a classification of extremal double circulant self-dual codes of length \(14\) is given. We present other double circulant codes which improve the lower bounds on the highest possible minimum weight. A classification of double circulant self-dual codes with parameters \([18, 9, 7]\) and \([24, 12, 9]\) is also given.