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 5X16302266&_mathId=si1.gif&_user=111111111&_pii=S0012365X16302266&_rdoc=1&_issn=0012365X&md5=da57fa852c376eb96664da169f3694b1" title="Click to view the MathML source">F_q${\mathbb{F}}_q$ be a finite field of odd order 5X16302266&_mathId=si2.gif&_user=111111111&_pii=S0012365X16302266&_rdoc=1&_issn=0012365X&md5=eaf0d28d3676b8ecfc88eeed84921e73" title="Click to view the MathML source">q$q$ and 5X16302266&_mathId=si3.gif&_user=111111111&_pii=S0012365X16302266&_rdoc=1&_issn=0012365X&md5=6711c8a9fb6c8b7c455a5b8c76db672b">5X16302266-si3.gif">$n=2^a{p}_1^{a_1}{p}_2^{a_2}$, where 5X16302266&_mathId=si4.gif&_user=111111111&_pii=S0012365X16302266&_rdoc=1&_issn=0012365X&md5=245cdfd9de929eb51b724b4cb5818f23" title="Click to view the MathML source">a,a₁,a₂$a,a_1,a_2$ are positive integers, 5" class="mathmlsrc">5X16302266&_mathId=si5.gif&_user=111111111&_pii=S0012365X16302266&_rdoc=1&_issn=0012365X&md5=225b590d5065e8b0e95baca67d211084" title="Click to view the MathML source">p₁,p₂5.gif">p1,p2 are distinct odd primes and 5X16302266&_mathId=si6.gif&_user=111111111&_pii=S0012365X16302266&_rdoc=1&_issn=0012365X&md5=022593412ab9528da844f0022a3ad925" title="Click to view the MathML source">4p₁p₂|q−1$4p_1{p}_2|q-1$. In this paper, we study the irreducible factorization of 5X16302266&_mathId=si7.gif&_user=111111111&_pii=S0012365X16302266&_rdoc=1&_issn=0012365X&md5=4ae1a00bdf5619edc70cc375a96e4701" title="Click to view the MathML source">xⁿ−1$x^n-1$ over 5X16302266&_mathId=si1.gif&_user=111111111&_pii=S0012365X16302266&_rdoc=1&_issn=0012365X&md5=da57fa852c376eb96664da169f3694b1" title="Click to view the MathML source">F_q${\mathbb{F}}_q$ and all primitive idempotents in the ring 5X16302266&_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〉${\mathbb{F}}_q\left[x\right]∕〈x^n-1〉$.Moreover, we obtain the dimensions and the minimum Hamming distances of all irreducible cyclic codes of length 5X16302266&_mathId=si10.gif&_user=111111111&_pii=S0012365X16302266&_rdoc=1&_issn=0012365X&md5=17c06ff9daf47d0fc0b0cbe420a44395" title="Click to view the MathML source">n$n$ over 5X16302266&_mathId=si1.gif&_user=111111111&_pii=S0012365X16302266&_rdoc=1&_issn=0012365X&md5=da57fa852c376eb96664da169f3694b1" title="Click to view the MathML source">F_q${\mathbb{F}}_q$.