Minimum distances of binary optimal LCD codes of dimension five are completely determined

Yang Liu; Ruihu Li; Qiang Fu; Hao Song

doi:10.3934/math.2024933

paper

Minimum distances of binary optimal LCD codes of dimension five are completely determined

arXiv:2210.05238 · doi:10.3934/math.2024933

Abstract

Let $t \in {2, 8, 10, 12, 14, 16, 18}$ and $n = 31 s + t \geq 14$ , $d_{a} (n, 5)$ and $d_{l} (n, 5)$ be distances of binary $[n, 5]$ optimal linear codes and optimal linear complementary dual (LCD) codes, respectively. We show that an $[n, 5, d_{a} (n, 5)]$ optimal linear code is not an LCD code, there is an $[n, 5, d_{l} (n, 5)] = [n, 5, d_{a} (n, 5) - 1]$ optimal LCD code if $t \neq = 16$ , and an optimal $[n, 5, d_{l} (n, 5)]$ optimal LCD code has $d_{l} (n, 5) = 16 s + 6 = d_{a} (n, 5) - 2$ for $t = 16$ . Combined with known results on optimal LCD code, $d_{l} (n, 5)$ of all $[n, 5]$ LCD codes are completely determined.

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Cited by in corpus (1)

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