Several families of q-ary cyclic codes with length \(q^m-1\)

It is very hard to construct an infinite family of cyclic codes of rate close to one half whose minimum distances have a good bound. Tang-Ding codes are very interesting, as their minimum distances have a square-root-like bound. Recently, a new generalization of Tang-Ding codes has been presented, Sun constructed several infinite families of binary cyclic codes with length \(2^{m}-1\) $2^m-1$ and dimension near \(2^{m-1}\) $2^{m-1}$ whose minimum distances much exceed the square-root bound (Sun, Finite Fields Appl. 89, 102200, 2023). In this paper, we construct several families of q-ary cyclic codes with length \(q^{m}-1\) $q^m-1$ and dimension near \(\frac{q^{m}-1}{2}\) $\frac{q^m-1}{2}$ , where \(q\ge 3\) $q\ge 3$ is a prime power and \(m \ge 3\) $m\ge 3$ is an integer. The minimum distances of these codes and their dual codes much exceed the square-root bound.