Several new classes of optimal ternary cyclic codes with two or three zeros

Abstract

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 \(\alpha \) be a generator of \(\mathbb F_{3^m}\setminus \{0\}\), where m is a positive integer. Denote by \(\mathcal {C}_{(i_1,i_2,\cdots , i_t)}\) the cyclic code with generator polynomial \(m_{\alpha ^{i_1}}(x)m_{\alpha ^{i_2}}(x)\cdots m_{\alpha ^{i_t}}(x)\), where \({{m}_{\alpha ^{i}}}(x)\) is the minimal polynomial of \({{\alpha }^{i}}\) over \({{\mathbb {F}}_{3}}\). In this paper, by analyzing the solutions of certain equations over finite fields, we present four classes of optimal ternary cyclic codes \(\mathcal {C}_{(0,1,e)}\) and \(\mathcal {C}_{(1,e,s)}\) with parameters \([3^m-1,3^m-\frac{3m}{2}-2,4]\), where \(s=\frac{3^m-1}{2}\). In addition, by determining the solutions of certain equations and analyzing the irreducible factors of certain polynomials over \(\mathbb F_{3^m}\), we present four classes of optimal ternary cyclic codes \(\mathcal {C}_{(2,e)}\) and \(\mathcal {C}_{(1,e)}\) with parameters \([3^m-1,3^m-2m-1,4]\). We show that our new optimal cyclic codes are not covered by known ones.