Several classes of optimal \(p\)-ary cyclic codes with minimum distance four. (English) Zbl 1542.94246
Summary: Cyclic codes are a subclass of linear codes and have wide applications in data storage systems, communication systems and consumer electronics due to their efficient encoding and decoding algorithms. Let \(p \geq 5\) be an odd prime and \(m\) be a positive integer. Let \(\mathcal{C}_{(1, e, s)}\) denote the \(p\)-ary cyclic code with three nonzeros \(\alpha, \alpha^e\), and \(\alpha^s\), where \(\alpha\) is a generator of \(\mathbb{F}_{p^m}^{\ast}, s = \frac{p^m -1}{2}\), and \(2 \leq e \leq p^m -2\). In this paper, by analyzing the solutions of certain equations over finite fields, we present four classes of optimal \(p\)-ary cyclic codes \(\mathcal{C}_{(1, e, s)}\) with parameters \([p^m -1, p^m -2m-2, 4]\). Some known results on optimal quinary cyclic codes with parameters \([5^m -1, 5^m -2m -2, 4]\) are special cases of our constructions. In addition, by analyzing the irreducible factors of certain polynomials over \(\mathbb{F}_{5^m}\), we present two classes of optimal quinary cyclic codes \(\mathcal{C}_{(1, e, s)}\).
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