On the minimum weights of binary LCD codes and ternary LCD codes,Finite Fields and Their Applications

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On the minimum weights of binary LCD codes and ternary LCD codes
Finite Fields and Their Applications ( IF 1.2 ) Pub Date : 2021-09-20 , DOI: 10.1016/j.ffa.2021.101925
Makoto Araya ₁ , Masaaki Harada ₂ , Ken Saito ₂

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Linear complementary dual (LCD) codes are linear codes that intersect with their dual codes trivially. We study the largest minimum weight $d_{2} (n, k)$ among all binary LCD $[n, k]$ codes and the largest minimum weight $d_{3} (n, k)$ among all ternary LCD $[n, k]$ codes. The largest minimum weights $d_{2} (n, 5)$ and $d_{3} (n, 4)$ are partially determined. We also determine the largest minimum weights $d_{2} (n, n - 5)$ , $d_{3} (n, n - i)$ for $i \in {2, 3, 4}$ , and $d_{3} (n, k)$ for $n \in {11, 12, \dots, 19}$ .

中文翻译：

二进制LCD码和三进制LCD码的最小权重

线性互补对偶 (LCD) 码是与对偶码简单相交的线性码。我们研究最大的最小权重 $d_{2} (n, 克)$ 在所有二进制 LCD 中 $[n, 克]$ 代码和最大最小权重 $d_{3} (n, 克)$ 在所有三元液晶显示器中 $[n, 克]$ 代码。最大最小权重 $d_{2} (n, 5)$ 和 $d_{3} (n, 4)$ 部分确定。我们还确定了最大的最小权重 $d_{2} (n, n - 5)$ , $d_{3} (n, n - 一世)$ 为了 $一世 \in {2, 3, 4}$ ，和 $d_{3} (n, 克)$ 为了 $n \in {11, 12, \dots, 19}$ .

更新日期：2021-09-21

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