A class of minimal cyclic codes over finite fields

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• 作者：Fen Li^a ; ^{lifen08001@163.com" class="auth_mail" title="E-mail the corresponding author} ; Xiwang Cao^a ; ^b ; ^{xwcao@nuaa.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Irreducible factorization ; Primitive idempotent ; Irreducible cyclic code
• 刊名：Discrete Mathematics
• 出版年：2017
• 出版时间：6 January 2017
• 年：2017
• 卷：340
• 期：1
• 页码：3197-3206
• 全文大小：435 K

文摘

Let F_q${\mathbb{F}}_q$ be a finite field of odd order q$q$ and $n=2^a{p}_1^{a_1}{p}_2^{a_2}$, where a,a₁,a₂$a,a_1,a_2$ are positive integers, p₁,p₂$p_1,p_2$ are distinct odd primes and 4p₁p₂|q−1$4p_1{p}_2|q-1$. In this paper, we study the irreducible factorization of xⁿ−1$x^n-1$ over F_q${\mathbb{F}}_q$ and all primitive idempotents in the ring F_q[x]∕〈xⁿ−1〉${\mathbb{F}}_q\left[x\right]∕〈x^n-1〉$.Moreover, we obtain the dimensions and the minimum Hamming distances of all irreducible cyclic codes of length n$n$ over F_q${\mathbb{F}}_q$.