Several families of q-ary cyclic codes with length $$q^m-1$$

Abstract

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 \(2^{m}-1\) and dimension near \(2^{m-1}\) 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 \(q^{m}-1\) and dimension near \(\frac{q^{m}-1}{2}\), where \(q\ge 3\) is a prime power and \(m \ge 3\) is an integer. The minimum distances of these codes and their dual codes much exceed the square-root bound.