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arXiv · 1701.04165

Some Bounds on Binary LCD Codes

Lucky Galvez·Jon-Lark Kim·Nari Lee·Young Gun Roe·Byung-Sun Won
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Abstract

A linear code with a complementary dual (or LCD code) is defined to be a linear code $C$ whose dual code $C^{\perp}$ satisfies $C \cap C^{\perp}$= $\left\{ \mathbf{0}\right\} $. Let $LCD{[}n,k{]}$ denote the maximum of possible values of $d$ among $[n,k,d]$ binary LCD codes. We give exact values of $LCD{[}n,k{]}$ for $1 \le k \le n \le 12$. We also show that $LCD[n,n-i]=2$ for any $i\geq2$ and $n\geq2^{i}$. Furthermore, we show that $LCD[n,k]\leq LCD[n,k-1]$ for $k$ odd and $LCD[n,k]\leq LCD[n,k-2]$ for $k$ even.

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Lucky Galvez, Jon-Lark Kim, Nari Lee, Young Gun Roe, Byung-Sun Won. 2017-01-16. Some Bounds on Binary LCD Codes. https://arxiv.org/abs/1701.04165

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