Double circulant codes improve binary LCD error correction rates
Double circulant LCD codes over $\mathbb{F}_2+u\mathbb{F}_2$ and constructions for binary LCD codes
Information Theory
Summary
Some codes used in computers and communications need to be easy to check and fix errors. The authors look at a special kind of code built from repeating patterns called double circulant codes over a simple mathematical ring. They figured out exactly how many such codes exist and showed how to make good binary codes that can correct errors well. They also made specific examples of these codes that have strong error correction with neat properties useful for practical use.
What this means in practice
- •For communication engineers: Design binary error-correcting codes with guaranteed good rates and distances for reliable data transmission.
- •For hardware designers: Construct efficient and secure error-correcting code blocks using explicitly enumerated double circulant LCD codes.
Authors
Xingxing Xu, Minjia Shi, Jon-Lark Kim
Abstract
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 $1/2$ whose relative distance can approach $H_2^{-1}(1/4)$ from below. More importantly from the constructive viewpoint, by combining Gray images with a known extension construction, we obtain a binary LCD code with parameters $[29,15,7]$ and construct two explicit permutation-inequivalent binary LCD codes with parameters $[45,23,9]$.