A class of optimal ternary cyclic codes and their duals

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• 作者：Cuiling Fan^a ; ^{cuilingfan@163.com" class="auth_mail" title="E-mail the corresponding author} ; Nian Li^b ; ^{nianli.2010@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Zhengchun Zhou^a ; ^{zzc@home.swjtu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; ^{zczhou@126.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：94B15 ; 11T71
• 刊名：Finite Fields and Their Applications
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：37
• 期：Complete
• 页码：193-202
• 全文大小：303 K

文摘

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. Let m=2ℓ+1$m=2\ell +1$ for an integer ℓ≥1$\ell \ge 1$ and π   be a generator of GF(3^m)^⁎$\mathbb{GF}{(3^m)}^{⁎}$. In this paper, a class of cyclic codes C_(u,v)${\mathcal{C}}_{(u,v)}$ over GF(3)$\mathbb{GF}(3)$ with two nonzeros π^u${\pi }^u$ and 94b40bdde3556ca810bde" title="Click to view the MathML source">π^v${\pi }^v$ is studied, where u=(3^m+1)/2$u=(3^m+1)/2$, and v=2⋅3^ℓ+1$v=2\cdot 3^{\ell }+1$ is the ternary Welch-type exponent. Based on a result on the non-existence of solutions to certain equation over GF(3^m)$\mathbb{GF}(3^m)$, the cyclic code C_(u,v)${\mathcal{C}}_{(u,v)}$ is shown to have minimal distance four, which is the best minimal distance for any linear code over GF(3)$\mathbb{GF}(3)$ with length 3^m−1$3^m-1$ and dimension 3^m−1−2m$3^m-1-2m$ according to the Sphere Packing bound. The duals of this class of cyclic codes are also studied.