Explicit Cyclic and Quasi-Cyclic Codes With Optimal, Best Known Parameters, and Large Relative Minimum Distances

In this paper, we construct many infinite families of distance-optimal codes with new parameters, some of which are BCH codes and quasi-cyclic codes. In particular, we report the first infinite family of binary distance-optimal BCH codes with the minimum distance 8. Secondly, several infinite families of binary BCH codes and quasi-cyclic codes are presented. Many codes in these families have optimal or best known parameters. Thirdly, we construct infinite families of binary cyclic $\left [{{n, \geq \frac {n+1}{2},d}}\right]_{2}$ codes with minimum distances $d \geq \lceil \frac {n-1}{\prod _{i=1}^{s}p_{i}}\rceil $ , $n=(2^{p_{1}}-1)(2^{p_{2}}-1) \cdots (2^{p_{s}}-1)$ , $p_{1}, \ldots, p_{s}$ are different primes. Our construction extends the main result of a recent paper published by Sun et al. to much more general binary cyclic codes with various lengths. We also construct an infinite family of binary quasi-cyclic codes with the rate around $\frac {1}{2}$ and relative minimum distance lower bounded by $O\left ({{\frac {1}{\log _{2} \log _{2} n}}}\right)$ .