Double circulant LCD codes over $\mathbb{F}_2+u\mathbb{F}_2$ and their related binary LCD codes
Xingxing Xu, Bo Wu, Jon-Lark Kim
原文摘要
Linear complementary dual (LCD) codes are linear codes that intersect their dual trivially. We study double circulant (Euclidean) LCD codes over the finite chain ring $R=\mathbb F_2+u\mathbb F_2$, where $u^2=0$. Using the Chinese Remainder Theorem and the factorization of $x^n-1$, we derive an explicit enumeration formula, with a separate treatment of the constituent corresponding to $x-1$, which is not obtained by a formal specialization of the previously known counting formula over $\mathbb F_q[u]/(u^k)$. Assuming Artin's primitive root conjecture for $2$, we show that the binary Gray images contain an asymptotically good family of rate $1/2$ which attains the asymptotic Gilbert--Varshamov bound, i.e., whose relative distance can approach $H_2^{-1}(1/2)$ from below. More importantly from the constructive viewpoint, by combining Gray images with a known extension construction, we obtain a binary LCD code with previously unknown parameters $[29,15,7]$ and construct two explicit permutation-inequivalent binary LCD codes with parameters $[45,23,9]$ and nine binary LCD codes with the best known parameters.
cs.IT · math.RA