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 c376eb96664da169f3694b1" title="Click to view the MathML source">F_qmrow>Fmrow><mrow>qmrow> be a finite field of odd order q$q$ and mrow>2mrow><mrow>amrow><mrow>pmrow><mrow>1mrow><mrow><mrow>amrow><mrow>1mrow>mrow><mrow>pmrow><mrow>2mrow><mrow><mrow>amrow><mrow>2mrow>mrow>, where a,a₁,a₂mrow>amrow><mrow>1mrow>,<mrow>amrow><mrow>2mrow> are positive integers, p₁,p₂mrow>pmrow><mrow>1mrow>,<mrow>pmrow><mrow>2mrow> are distinct odd primes and 4p₁p₂|q−1mrow>pmrow><mrow>1mrow><mrow>pmrow><mrow>2mrow>|q−1. In this paper, we study the irreducible factorization of c375a96e4701" title="Click to view the MathML source">xⁿ−1mrow>xmrow><mrow>nmrow>−1 over c376eb96664da169f3694b1" title="Click to view the MathML source">F_qmrow>Fmrow><mrow>qmrow> and all primitive idempotents in the ring c302b903d50cdcc8e1cd9a255aab9931" title="Click to view the MathML source">F_q[x]∕〈xⁿ−1〉mrow>Fmrow><mrow>qmrow><mrow>[x]mrow>∕<mrow>〈<mrow>xmrow><mrow>nmrow>−1〉mrow>.Moreover, we obtain the dimensions and the minimum Hamming distances of all irreducible cyclic codes of length n$n$ over c376eb96664da169f3694b1" title="Click to view the MathML source">F_qmrow>Fmrow><mrow>qmrow>.