AI 中文总结
研究针对偶数$s$构造速率约为$1/2$的$[2^{s}-1,k]$二元循环码，使其最小距离$d$和对偶距离$d^\perp$同时很大，给出多种构造方法及参数，所构多数码最小距离更大，还获优异参数码并经验证。
AI 中文摘要
二元循环码是一类重要的线性码，具有高效的编码和解码算法。几十年来，构造具有良好参数的二元循环码一直是一个活跃的研究课题。近年来，在研究$k$接近$n/2$且$d\geq \sqrt{n}$的二元$[n,k,d]$循环码方面取得了重大进展。然而，在已知的此类码中，只有少数族同时具有大的最小距离和大的对偶距离。本文针对偶数$s$，专注于构造速率约为$1/2$的$[2^{s}-1,k]$二元循环码，其最小距离$d$和对偶距离$d^\perp$同时很大。一些构造产生参数为$[2^s-1,2^{s-1}\pm 1,d]$的二元循环码，使得$d\cdot d^\perp$的下界接近$n$或$2n$。其他构造产生参数为$[2^s -1, 2^{s-1} + c, d]$（其中$1- \frac{5s}{2} \leq c \leq \frac{5s}{2} -1$）的二元循环码，使得$d\cdot d^\perp$的下界接近$n$或$2n$。与已知的相同长度和维度的二元循环码相比，我们构造的大多数二元循环码具有更大的最小距离。特别是，我们获得了几个具有优异参数的码，并根据此http URL处的码表进行了验证。
英文摘要
Binary cyclic codes are an important class of linear codes that admit highly efficient encoding and decoding algorithms. Constructing binary cyclic codes with favorable parameters has been an active research topic for several decades. In recent years, substantial progress has been made in the study of binary $[n,k,d]$ cyclic codes with $k$ close to $n/2$ and $d\geq \sqrt{n}$. Nevertheless, among the known codes of this type, only a few families simultaneously possess both large minimum distances and large dual distances. In this paper, for even $s$, we focus on new constructions of $[2^{s}-1,k]$ binary cyclic codes with rate approximately $1/2$, for which both the minimum distance $d$ and the dual distance $d^\perp$ are simultaneously large. Some of the constructions yield binary cyclic codes with parameters $[2^s-1,2^{s-1}\pm 1,d]$ such that the lower bounds of $d\cdot d^\perp$ are close to $n$ or $2n$. The other constructions yield binary cyclic codes with parameters $[2^s -1, 2^{s-1} + c, d]$ for some $ 1- \frac{5s}{2} \leq c \leq \frac{5s}{2} -1$ such that the lower bounds of $d\cdot d^\perp$ are close to $n$ or $2n$. Compared with the known binary cyclic codes of the same length and dimension, the majority of the binary cyclic codes we construct exhibit larger minimum distances. In particular, we obtain several codes with excellent parameters, which are verified against the Code Tables available at http://www.codetables.de/.
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