Cyclic and Negacyclic Codes With Optimal and Best Known Minimum Distances

In this paper, we construct infinitely many families of distance-optimal binary BCH codes with the minimum distance 6 and an infinite family of distance-optimal quaternary BCH codes with the minimum distance 4. We also construct several infinite families of cyclic and negacyclic BCH codes over ${\mathbf { F}}_{2}$ , ${\mathbf { F}}_{3}$ , ${\mathbf { F}}_{4}$ , ${\mathbf { F}}_{5}$ , ${\mathbf { F}}_{7}$ and ${\mathbf { F}}_{9}$ with good parameters $n,\,k,\,d$ , such that the maximal possible minimum distance $d_{\max }$ of a linear $[n, k]_{q}$ code is at most $d_{\max } \leq d+8$ . Many codes in these families have optimal or best known minimum distances. 145 optimal or best known codes are constructed as cyclic codes, negacyclic codes, their shortening codes and punctured codes. Several infinite families of rate $\frac {1}{2}$ negacyclic $\left [{{n, \frac {n+1}{2}, d}}\right]_{q}$ codes or $\left [{{n, \frac {n}{2}, d}}\right]_{q}$ codes, such that their minimum distances satisfy $d\geq \frac {cn}{\log _{q} n}$ , where c is a positive constant, are also constructed. These are first several families of such negacyclic codes reported in the literature.