Repeated-root constacyclic codes of length 3lp^s and their dual codes

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• 作者：Li Liu ^{liuli-1128@163.com" class="auth_mail" title="E-mail the corresponding author} ; Lanqiang Li ; ^{lilanqiang716@126.com" class="auth_mail" title="E-mail the corresponding author} ; Xiaoshan Kai ; Shixin Zhu
• 关键词：94B05 ; 11T71
• 刊名：Finite Fields and Their Applications
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：42
• 期：Complete
• 页码：269-295
• 全文大小：497 K

文摘

Let edaa99c3c9c774137c8969c4ae2" title="Click to view the MathML source">p≠3$p\ne 3$ be any prime and l≠3$l\ne 3$ be any odd prime with gcd⁡(p,l)=1$\gcd (p,l)=1$. The multiplicative group $F_q^{⁎}=〈\xi 〉$ can be decomposed into mutually disjoint union of gcd⁡(q−1,3lp^s)$\gcd (q-1,3l{p}^s)$ cosets over the subgroup 〈ξ^3lps〉$〈{\xi }^{3l{p}^s}〉$, where ξ   is a primitive (q−1)$(q-1)$th root of unity. We classify all repeated-root constacyclic codes of length 3lp^s$3l{p}^s$ over the finite field F_q$F_q$ into some equivalence classes by this decomposition, where q=p^m$q=p^m$, s and m   are positive integers. According to these equivalence classes, we explicitly determine the generator polynomials of all repeated-root constacyclic codes of length 3lp^s$3l{p}^s$ over F_q$F_q$ and their dual codes. Self-dual cyclic codes of length 3lp^s$3l{p}^s$ over F_q$F_q$ exist only when p=2$p=2$. We give all self-dual cyclic codes of length 3⋅2^sl$3\cdot 2^{s}l$ over F_2^m$F_{2^m}$ and their enumeration. We also determine the minimum Hamming distance of these codes when gcd⁡(3,q−1)=1$\gcd (3,q-1)=1$ and l=1$l=1$.