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 i1" class="mathmlsrc">i1.gif&_user=111111111&_pii=S1071579715001057&_rdoc=1&_issn=10715797&md5=bc03d78747b52e918da3a2561a2e53aa" title="Click to view the MathML source">m=2ℓ+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 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 i10" class="mathmlsrc">i10.gif&_user=111111111&_pii=S1071579715001057&_rdoc=1&_issn=10715797&md5=0d766b43c875d891e05793aa3443dee0" title="Click to view the MathML source">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 i11" class="mathmlsrc">i11.gif&_user=111111111&_pii=S1071579715001057&_rdoc=1&_issn=10715797&md5=d3ae54f5577a97c0cb289e7eb774c126" title="Click to view the MathML source">3^m−1 and dimension i12" class="mathmlsrc">i12.gif&_user=111111111&_pii=S1071579715001057&_rdoc=1&_issn=10715797&md5=8747b6535b1075b5047bdff32086607a" title="Click to view the MathML source">3^m−1−2m according to the Sphere Packing bound. The duals of this class of cyclic codes are also studied.