Two classes of new optimal ternary cyclic codes

Abstract

<p style='text-indent:20px;'>Due to their wide applications in consumer electronics, data storage systems and communication systems, cyclic codes have been an interesting subject of study in recent years. The construction of optimal cyclic codes over finite fields is important as they have maximal minimum distance once the length and dimension are given. In this paper, we present two classes of new optimal ternary cyclic codes <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{C}_{(2,v)} $\end{document}</tex-math></inline-formula> by using monomials <inline-formula><tex-math id="M2">\begin{document}$ x^2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ x^v $\end{document}</tex-math></inline-formula> for some suitable <inline-formula><tex-math id="M4">\begin{document}$ v $\end{document}</tex-math></inline-formula> and explain the novelty of the codes. Furthermore, the weight distribution of <inline-formula><tex-math id="M5">\begin{document}$ \mathcal{C}_{(2,v)}^{\perp} $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M6">\begin{document}$ v = \frac{3^{m}-1}{2}+2(3^{k}+1) $\end{document}</tex-math></inline-formula> is determined.</p>