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

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Let class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302266&_mathId=si1.gif&_user=111111111&_pii=S0012365X16302266&_rdoc=1&_issn=0012365X&md5=da57fa852c376eb96664da169f3694b1" title="Click to view the MathML source">F_qclass="mathContainer hidden">class="mathCode">${\mathbb{F}}_q$ be a finite field of odd order class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302266&_mathId=si2.gif&_user=111111111&_pii=S0012365X16302266&_rdoc=1&_issn=0012365X&md5=eaf0d28d3676b8ecfc88eeed84921e73" title="Click to view the MathML source">qclass="mathContainer hidden">class="mathCode">$q$ and class="mathmlsrc">class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302266&_mathId=si3.gif&_user=111111111&_pii=S0012365X16302266&_rdoc=1&_issn=0012365X&md5=6711c8a9fb6c8b7c455a5b8c76db672b">class="imgLazyJSB inlineImage" height="19" width="77" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0012365X16302266-si3.gif">class="mathContainer hidden">class="mathCode">$n=2^a{p}_1^{a_1}{p}_2^{a_2}$, where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302266&_mathId=si4.gif&_user=111111111&_pii=S0012365X16302266&_rdoc=1&_issn=0012365X&md5=245cdfd9de929eb51b724b4cb5818f23" title="Click to view the MathML source">a,a₁,a₂class="mathContainer hidden">class="mathCode">$a,a_1,a_2$ are positive integers, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302266&_mathId=si5.gif&_user=111111111&_pii=S0012365X16302266&_rdoc=1&_issn=0012365X&md5=225b590d5065e8b0e95baca67d211084" title="Click to view the MathML source">p₁,p₂class="mathContainer hidden">class="mathCode">$p_1,p_2$ are distinct odd primes and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302266&_mathId=si6.gif&_user=111111111&_pii=S0012365X16302266&_rdoc=1&_issn=0012365X&md5=022593412ab9528da844f0022a3ad925" title="Click to view the MathML source">4p₁p₂|q−1class="mathContainer hidden">class="mathCode">$4p_1{p}_2|q-1$. In this paper, we study the irreducible factorization of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302266&_mathId=si7.gif&_user=111111111&_pii=S0012365X16302266&_rdoc=1&_issn=0012365X&md5=4ae1a00bdf5619edc70cc375a96e4701" title="Click to view the MathML source">xⁿ−1class="mathContainer hidden">class="mathCode">$x^n-1$ over class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302266&_mathId=si1.gif&_user=111111111&_pii=S0012365X16302266&_rdoc=1&_issn=0012365X&md5=da57fa852c376eb96664da169f3694b1" title="Click to view the MathML source">F_qclass="mathContainer hidden">class="mathCode">${\mathbb{F}}_q$ and all primitive idempotents in the ring class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302266&_mathId=si9.gif&_user=111111111&_pii=S0012365X16302266&_rdoc=1&_issn=0012365X&md5=c302b903d50cdcc8e1cd9a255aab9931" title="Click to view the MathML source">F_q[x]∕〈xⁿ−1〉class="mathContainer hidden">class="mathCode">${\mathbb{F}}_q\left[x\right]∕〈x^n-1〉$.Moreover, we obtain the dimensions and the minimum Hamming distances of all irreducible cyclic codes of length class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302266&_mathId=si10.gif&_user=111111111&_pii=S0012365X16302266&_rdoc=1&_issn=0012365X&md5=17c06ff9daf47d0fc0b0cbe420a44395" title="Click to view the MathML source">nclass="mathContainer hidden">class="mathCode">$n$ over class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302266&_mathId=si1.gif&_user=111111111&_pii=S0012365X16302266&_rdoc=1&_issn=0012365X&md5=da57fa852c376eb96664da169f3694b1" title="Click to view the MathML source">F_qclass="mathContainer hidden">class="mathCode">${\mathbb{F}}_q$.