Abstract
It has been shown by Bogdanova and Boukliev [1] that there exist a ternary [38,5,24] code and a ternary [37,5,23] code. But it is unknown whether or not there exist a ternary [39,6,24] code and a ternary [38,6,23] code. The purpose of this paper is to prove that (1) there is no ternary [39,6,24] code and (2) there is no ternary [38,6,23] code using the nonexistence of ternary [39,6,24] codes. Since it is known (cf. Brouwer and Sloane [2] and Hamada and Watamori [14]) that (i) n3(6,23) = 38> or 39 and d3(38,6) = 22 or 23 and (ii) n3(6,24) = 39 or 40 and d3(39,6) = 23 or 24, this implies that n3(6,23) = 39, d3(38,6) = 22, n3(6,24) = 40 and d3(39,6) = 23, where n3<>(k,d) and d<>3(n,k) denote the smallest value of n and the largest value of d, respectively, for which there exists an [n,k,d] code over the Galois field GF(3).
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Hamada, N., Van Eupen, M. The Nonexistence of Ternary [38, 6, 23] Codes. Designs, Codes and Cryptography 13, 165–172 (1998). https://doi.org/10.1023/A:1008278312966
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DOI: https://doi.org/10.1023/A:1008278312966