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>

Two Classes of Optimal Ternary Cyclic Codes with Minimum Distance Four

“…Ding and Helleseth [7] constructed several classes of optimal ternary cyclic codes C (1,e) with parameters [3 m − 1, 3 m − 1 − 2m, 4] by using almost perfect nonlinear monomials and some other monomials over F 3 m . Subsequently, many classes of optimal ternary cyclic codes C (1,e) with parameters [3 m − 1, 3 m − 1 − 2m, 4] were constructed successively [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22].…”

A Class of New Optimal Ternary Cyclic Codes over F<sub>3<sup><i>m</i></sup></sub> with Minimum Distance 4

“…They also discussed the weight distribution of the dual of this code. In 2020, Liu, Cao and Lu [6] studied the code ( )…”

Two Families of Optimal Ternary Cyclic Codes With Two Zeros

“…After that, many classes of optimal ternary cyclic codes C 3 (1, v) were constructed in [4]- [11] and [14]. For the case u = 1, four classes of optimal ternary cyclic codes C 3 (u, v) were investigated in [9], [10], [17] and [18], where…”