Construction of Some Optimal Ternary Linear Codes and the Uniqueness of [294, 6, 195; 3]-Codes Meeting the Griesmer Bound

Let nq(k, d) denote the smallest value of n for which there exists an [n, k, d; q]-code. It is known (cf. (J. Combin. Inform. Syst. Sci.18, 1993, 161–191)) that (1) n3(6, 195) {294, 295}, n3(6, 194) {293, 294}, n3(6, 193) {292, 293}, n3(6, 192) {290, 291}, n3(6, 191) {289, 290}, n3(6, 165) {250, 251} and (2) there is a one-to-one correspondence between the set of all nonequivalent [294, 6, 195; 3]-codes meeting the Griesmer bound and the set of all {v2 + 2v3 + v4, v1 + 2v2 + v3; 5, 3}-minihypers, where vi = (3i − 1)/(3 − 1) for any integer i ≥ 0. The purpose of this paper is to show that (1) n3(6, 195) = 294, n3(6, 194) = 293, n3(6, 193) = 292, n3(6, 192) = 290, n3(6, 191) = 289, n3(6, 165) = 250 and (2) a [294, 6, 195; 3]-code is unique up to equivalence using a characterization of the corresponding {v2 + 2v3 + v4, v1 + 2v2 + v3; 5, 3}-minihypers.