On a conjecture about a class of optimal ternary cyclic codes

Li, Nian; Zhou, Zhengchun; Helleseth, Tor

doi:10.1109/iwsda.2015.7458415

Cited by 26 publications

(11 citation statements)

References 18 publications

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“…and e is some type of integer, the cyclic codes C (1,e) with parameters [3 m − 1, 3 m − 1 − 2m, 4] have been studied [2,16,[22][23][24][25] ).…”

Section: Preliminariesmentioning

confidence: 99%

“…Moreover, it is a worthwhile subject to construct optimal cyclic codes over finite fields as they have the best error correcting capability among all codes with the same length and dimension. In recent years, many optimal ternary cyclic codes [2,4,16,[22][23][24][25] have been constructed. For example, in Ref.…”

Section: Introductionmentioning

confidence: 99%

“…Since then, a few of the open problems were solved by Li et al in Refs. [23,24], by discussing the non-existence of solutions to a certain equation over F 3 m which has been mentioned in Ref. [22].…”

Section: Introductionmentioning

confidence: 99%

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Three Classes of Optimal Ternary Cyclic Codes and the Weight Distributions of Their Duals

Zhu

Liu

2019

Chinese Journal of Electronics

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“…and e is some type of integer, the cyclic codes C (1,e) with parameters [3 m − 1, 3 m − 1 − 2m, 4] have been studied [2,16,[22][23][24][25] ).…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Three Classes of Optimal Ternary Cyclic Codes and the Weight Distributions of Their Duals

Zhu

Liu

2019

Chinese Journal of Electronics

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“…In 2013, Ding and Helleseth [4] obtained some optimal ternary cyclic codes by employing almost perfect nonlinear monomials and some other monomials over F p m . They also presented nine open problems, some of which were solved by N. Li et al [13,14]. In 2014, Zhou and Ding [30] determined the weight distribution of a class of three-weight cyclic codes and studied the duals of a subclass of them which were proved to be optimal.…”

Section: Introductionmentioning

confidence: 99%

Two classes of new optimal ternary cyclic codes

Liu¹,

Cao²,

Lu³

2023

AMC

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<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>

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“…They also presented nine open problems. After that, two of the open problems were solved by N. Li et al [13,14]. In [13], the authors also presented several classes of optimal cyclic codes with parametes…”

Section: Introductionmentioning

confidence: 99%

Optimal p-ary cyclic codes with two zeros

Liu

Cao

2019

Preprint

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As a subclass of linear codes, cyclic codes have efficient encoding and decoding algorithms, so they are widely used in many areas such as consumer electronics, data storage systems and communication systems. In this paper, we give a general construction of optimal p-ary cyclic codes which leads to three explicit constructions. In addition, another class of p-ary optimal cyclic codes are presented.

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On a conjecture about a class of optimal ternary cyclic codes

Cited by 26 publications

References 18 publications

Three Classes of Optimal Ternary Cyclic Codes and the Weight Distributions of Their Duals

Three Classes of Optimal Ternary Cyclic Codes and the Weight Distributions of Their Duals

Two classes of new optimal ternary cyclic codes

Optimal p-ary cyclic codes with two zeros

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