Binary Cyclic Codes With Simultaneously Large Minimum Distances and Dual Distances

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/.