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 pan id="mmlsi1" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300557&_mathId=si1.gif&_user=111111111&_pii=S1071579716300557&_rdoc=1&_issn=10715797&md5=54945edaa99c3c9c774137c8969c4ae2" title="Click to view the MathML source">p≠3pan>pan class="mathContainer hidden">pan class="mathCode">$p\ne 3$pan>pan>pan> be any prime and pan id="mmlsi2" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300557&_mathId=si2.gif&_user=111111111&_pii=S1071579716300557&_rdoc=1&_issn=10715797&md5=180200901d206332322fc490385d7c05" title="Click to view the MathML source">l≠3pan>pan class="mathContainer hidden">pan class="mathCode">$l\ne 3$pan>pan>pan> be any odd prime with pan id="mmlsi3" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300557&_mathId=si3.gif&_user=111111111&_pii=S1071579716300557&_rdoc=1&_issn=10715797&md5=bf839aad6af8765fb64bdf3bfb107e89" title="Click to view the MathML source">gcd⁡(p,l)=1pan>pan class="mathContainer hidden">pan class="mathCode">$\gcd (p,l)=1$pan>pan>pan>. The multiplicative group pan id="mmlsi156" class="mathmlsrc">pan class="mathContainer hidden">pan class="mathCode">$F_q^{⁎}=〈\xi 〉$pan>pan>pan> can be decomposed into mutually disjoint union of pan id="mmlsi5" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300557&_mathId=si5.gif&_user=111111111&_pii=S1071579716300557&_rdoc=1&_issn=10715797&md5=ebb39a1d7f0001ff1f8b10858fbb5672" title="Click to view the MathML source">gcd⁡(q−1,3lp^s)pan>pan class="mathContainer hidden">pan class="mathCode">$\gcd (q-1,3l{p}^s)$pan>pan>pan> cosets over the subgroup pan id="mmlsi161" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300557&_mathId=si161.gif&_user=111111111&_pii=S1071579716300557&_rdoc=1&_issn=10715797&md5=27d3633c823f72c126bcf205005cd78b" title="Click to view the MathML source">〈ξ^3lps〉pan>pan class="mathContainer hidden">pan class="mathCode">$〈{\xi }^{3l{p}^s}〉$pan>pan>pan>, where ξ   is a primitive pan id="mmlsi7" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300557&_mathId=si7.gif&_user=111111111&_pii=S1071579716300557&_rdoc=1&_issn=10715797&md5=752deb5c33c7666e2e8349b6ad4fe4d4" title="Click to view the MathML source">(q−1)pan>pan class="mathContainer hidden">pan class="mathCode">$(q-1)$pan>pan>pan>th root of unity. We classify all repeated-root constacyclic codes of length pan id="mmlsi205" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300557&_mathId=si205.gif&_user=111111111&_pii=S1071579716300557&_rdoc=1&_issn=10715797&md5=806da4f23872fddd1dc58f86b94f8fc2" title="Click to view the MathML source">3lp^span>pan class="mathContainer hidden">pan class="mathCode">$3l{p}^s$pan>pan>pan> over the finite field pan id="mmlsi137" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300557&_mathId=si137.gif&_user=111111111&_pii=S1071579716300557&_rdoc=1&_issn=10715797&md5=d0924b24bfbcae2135c92a8a9bfa3109" title="Click to view the MathML source">F_qpan>pan class="mathContainer hidden">pan class="mathCode">$F_q$pan>pan>pan> into some equivalence classes by this decomposition, where pan id="mmlsi10" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300557&_mathId=si10.gif&_user=111111111&_pii=S1071579716300557&_rdoc=1&_issn=10715797&md5=519bfa415ce425a8f1196851c44f7510" title="Click to view the MathML source">q=p^mpan>pan class="mathContainer hidden">pan class="mathCode">$q=p^m$pan>pan>pan>, 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 pan id="mmlsi205" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300557&_mathId=si205.gif&_user=111111111&_pii=S1071579716300557&_rdoc=1&_issn=10715797&md5=806da4f23872fddd1dc58f86b94f8fc2" title="Click to view the MathML source">3lp^span>pan class="mathContainer hidden">pan class="mathCode">$3l{p}^s$pan>pan>pan> over pan id="mmlsi137" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300557&_mathId=si137.gif&_user=111111111&_pii=S1071579716300557&_rdoc=1&_issn=10715797&md5=d0924b24bfbcae2135c92a8a9bfa3109" title="Click to view the MathML source">F_qpan>pan class="mathContainer hidden">pan class="mathCode">$F_q$pan>pan>pan> and their dual codes. Self-dual cyclic codes of length pan id="mmlsi205" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300557&_mathId=si205.gif&_user=111111111&_pii=S1071579716300557&_rdoc=1&_issn=10715797&md5=806da4f23872fddd1dc58f86b94f8fc2" title="Click to view the MathML source">3lp^span>pan class="mathContainer hidden">pan class="mathCode">$3l{p}^s$pan>pan>pan> over pan id="mmlsi137" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300557&_mathId=si137.gif&_user=111111111&_pii=S1071579716300557&_rdoc=1&_issn=10715797&md5=d0924b24bfbcae2135c92a8a9bfa3109" title="Click to view the MathML source">F_qpan>pan class="mathContainer hidden">pan class="mathCode">$F_q$pan>pan>pan> exist only when pan id="mmlsi11" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300557&_mathId=si11.gif&_user=111111111&_pii=S1071579716300557&_rdoc=1&_issn=10715797&md5=19309a7e01078ad1096ed7b523adea3d" title="Click to view the MathML source">p=2pan>pan class="mathContainer hidden">pan class="mathCode">$p=2$pan>pan>pan>. We give all self-dual cyclic codes of length pan id="mmlsi12" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300557&_mathId=si12.gif&_user=111111111&_pii=S1071579716300557&_rdoc=1&_issn=10715797&md5=80898f92adf32861cf83e27161b78655" title="Click to view the MathML source">3⋅2^slpan>pan class="mathContainer hidden">pan class="mathCode">$3\cdot 2^{s}l$pan>pan>pan> over pan id="mmlsi13" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300557&_mathId=si13.gif&_user=111111111&_pii=S1071579716300557&_rdoc=1&_issn=10715797&md5=c34a6046bf00f00cc52cc9739eb9d17e" title="Click to view the MathML source">F_2^mpan>pan class="mathContainer hidden">pan class="mathCode">$F_{2^m}$pan>pan>pan> and their enumeration. We also determine the minimum Hamming distance of these codes when pan id="mmlsi14" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300557&_mathId=si14.gif&_user=111111111&_pii=S1071579716300557&_rdoc=1&_issn=10715797&md5=fe57218bbe9d31e2a212782050d545d5" title="Click to view the MathML source">gcd⁡(3,q−1)=1pan>pan class="mathContainer hidden">pan class="mathCode">$\gcd (3,q-1)=1$pan>pan>pan> and pan id="mmlsi15" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300557&_mathId=si15.gif&_user=111111111&_pii=S1071579716300557&_rdoc=1&_issn=10715797&md5=6acc9a42960dd1c6d79b93249bca99bc" title="Click to view the MathML source">l=1pan>pan class="mathContainer hidden">pan class="mathCode">$l=1$pan>pan>pan>.