环F_p+vF_p上自对偶码的构造（英文）
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摘要
In this paper, we give an explicit construction for self-dual codes over F_p+vF_p(v~2= v) and determine all the self-dual codes over F_p+ vF_p by using self-dual codes over finite field F_p, where p is a prime.
In this paper, we give an explicit construction for self-dual codes over F_p+vF_p(v~2= v) and determine all the self-dual codes over F_p+ vF_p by using self-dual codes over finite field F_p, where p is a prime.
In this paper, we give an explicit construction for self-dual codes over F_p+vF_p(v~2= v) and determine all the self-dual codes over F_p+ vF_p by using self-dual codes over finite field F_p, where p is a prime.
引文
[1]ALFARO R,DHUL-QARNAYN K.Constructing self-dual codes over Fq[u]/(ut)[J].Des Codes Cryptogr,2015,74(2):453-465.
[2]BLAKE I F.Codes over certain rings[J].Inform Control,1972,20:396-404.
[3]BLAKE I F.Codes over integer residue rings[J].Inform Control,1975,29:295-300.
[4]CHEN Bo-cong,LING San,ZHANG Guang-hui.Enumeration formulas for self-dualcyclic codes[J].Finite Fields Appl,2016,42:1-22.
[5]DOUGHERTY S T,FERN′ANDEZ-C′ORDOBA C.Z2Z4-additive formally self-dual codes[J].Des Codes Cryptogr,2014,72(2):435-453.
[6]DINH Hai Q,FAN Yun,LIU Hua-lu,LIU Xiu-sheng,SRIBOONCHITTA Songsak.On self-dual constacyclic codes of length ps over Fpm+uFpm[J].Discrete Math,2018,341(2):324-335.
[7]DINH H Q,L′OPEZ-PERMOUTH S R.Cyclic and negacyclic codes over finite chain rings[J].IEEE Trans Inform Theory,2004,50(8):1728-1744.
[8]DOUGHERTY S T,HARADA M,SOL′E P.Self-dual codes over rings and the Chinese remainder theorem[J].Hokkaido Math J,1999,28:253-283.
[9]DOUGHERTY Steven T,KIM Jon-Lark,KULOSMAN Hamid,LIU Hong-wei.Self-dualcodes over commutative Frobenius rings[J].Finite Fields Appl,2010,16:14-26.
[10]DOUGHERTY S T,KIM J-L,LIU H.Constructions of self-dual codes over finite commutative chain rings[J].Int J Inform Coding Theory,2010,1(2):171-190.
[11]DOUGHERTY Steven T,KAYA Abidin,SALT¨URK Esengl.Constructions of self-dualcodes and formally self-dual codes over rings[J].Appl Algebra Engrg Comm Comput,2016,27(5):435-449.
[12]DOUGHERTY Steven T,LEROY Andr,Euclidean self-dual codes over non-commutative Frobenius rings[J].Appl Algebra Engrg Comm Comput,2016,27(3):185-203.
[13]GILDEA Joe,KAYA Abidin,TAYLOR Rhian,YILDIZ Bahattin.Constructions for self-dual codes induced from group rings[J].Finite Fields Appl,2018,51:71-92.
[14]HAN Sunghyu.A method for constructing self-dual codes over Z2m[J].Des Codes Cryptogr,2015,75(2):253-262.
[15]HAMMONS A R,JR.,KUMAR P V,CALDERBANK A R,SLOANE N J A,SOL′E P.The Z4linearity of Kerdock,Preparata,Goethals and related codes[J].IEEE Trans Inform Theory,1994,40(2):301-319.
[16]JITMAN Somphong,LING San,SANGWISUT Ekkasit.On self-dual cyclic codes of length paover GR(p2,s)[J].Adv Math Commun,2016,10(2):255-273.
[17]KARADENIZ Suat,DOUGHERTY Steven T,YILDIZ Bahattin.Constructing formally self-dual codes over Rk[J].Discrete Appl Math,2014,167:188-196.
[18]KIM Boran,LEE Yoonjin.Lee weights of cyclic self-dual codes over Galois rings of characteristic p2[J].Finite Fields Appl,2017,45:107-130.
[19]KIM J-L,LEE Y.Euclidean and Hermitian self-dual MDS codes over large finite fields[J].J Combin Theory,Ser A,2004,105:79-95.
[20]KIM J-L,LEE Y.Construction of MDS self-dual codes over Galois rings[J].Des Codes Cryptogr,2007,45(2):247-258.
[21]KAYA Abidin,YILDIZ Bahattin.Various constructions for self-dual codes over rings and new binary self-dual codes[J].Discrete Math,2016,339(2):460-469.
[22]KAI Xiao-shan,ZHU Shi-xin.Negacyclic self-dual codes over finite chain rings[J].Des Codes Cryptogr,2012,62(2):161-174.
[23]KAI Xiao-shan,ZHU Shi-xin,TANG Yong-sheng.Some constacyclic self-dual codesover the integers modulo2m[J].Finite Fields Appl,2012,18(2):258-270.
[24]LING San,BLACKFORD J.Zpk+1-linear codes[J].IEEE Trans Inform Theory,2002,48:2592-2605.
[25]LIU Xiu-sheng,LIU Hua-lu.Cyclic codes over F2+uF2+vF2[J].Chin Quart J of Math,2014,29(2):189-194.
[26]LIU Xiu-sheng,LIU Hua-lu.The self-dual codes over formal power series rings[J].Chin Quart J of Math,2014,29(1):30-38.
[27]NAGATA Kiyoshi,NEMENZO Fidel,WADA Hideo.Mass formula and structure of self-dual codes over Z2s[J].Des Codes Cryptogr,2013,67(3):293-316.
[28]RAINS E,SLOANE N J A.Self-dual codes,in the Handbook of Coding Theory,V.S.Pless and W.C[J].Huffman,eds.,Elsevier,Amsterdam,1998,177-294.
[29]SHARMA Anuradha,SHARMA Amit K.Construction of self-dual codes over Z2m[J].Cryptogr Commun,2016,8(1):83-101.
[30]WOLFMANN J.Binary image of cyclic codes over Z4[J].IEEE Trans Inform Theory,2001,47(5):1773-1779.
[31]WOOD J.Duality for modules over finite rings and applications to coding theory[J].Amer J Math,1999,121:555-575.
[32]WU Yan-sheng,YUE Qin,LI Feng-wei.Primitive idempotents of irreducible cyclic codes and self-dual cyclic codes over Galois rings[J].Discrete Math,2018,341(6):1755-1767.
[33]YOO Jinjoo,LEE Yoonjin,KIM Boreum.Constructions of formally self-dual codesover Z4and their weight enumerators[J].IEEE Trans Inform Theory,2017,63(12):7667-7675.
[34]ZHANG Guang-hui.Constacyclic codes over Fp+vFp[J].Ars Comb,2018,139:55-67.
[35]ZHANG Guang-hui.A note on self-dual cyclic codes over finite chain rings[J].Adv Math(China),2016,45(2):206-210.
[36]ZHANG Guang-hui,LI Liang-chen.Self-dual codes and cyclic codes over Fp+vFp[J].Ars Comb,2014,116:445-455.
[37]ZHANG Guang-hui,LIU Hong-wei.Self-dual permutation codes over formal power series rings and finite principal ideal rings[J].Acta Math Sci Ser B Engl Ed,2013,33(6):1695-1710.
[38]ZHANG Guang-hui,LIU Qing-qing.Cyclic codes with complementary duals over Fp+vFp[J].Chin Quart J of Math,2016,31(2):118-124.
[39]ZHU Shi-xin,QIAN Kai-yan,KAI Xiao-shan.A note on negacyclic self-dual codesover Z2a[J].Discrete Math,2012,312(22):3270-3275.
[40]ZHU Shi-xin,WANG Li-qi.A class of constacyclic codes over Fp+vFp and its Gray image[J].Discrete Math,2011,311:2677-2682.
[41]ZHU Shi-xin,WANG Yu,SHI Min-jia.Some results on cyclic codes over F2+vF2[J].IEEE Trans Inform Theory,2010,56(4):1680-1684.
[2]BLAKE I F.Codes over certain rings[J].Inform Control,1972,20:396-404.
[3]BLAKE I F.Codes over integer residue rings[J].Inform Control,1975,29:295-300.
[4]CHEN Bo-cong,LING San,ZHANG Guang-hui.Enumeration formulas for self-dualcyclic codes[J].Finite Fields Appl,2016,42:1-22.
[5]DOUGHERTY S T,FERN′ANDEZ-C′ORDOBA C.Z2Z4-additive formally self-dual codes[J].Des Codes Cryptogr,2014,72(2):435-453.
[6]DINH Hai Q,FAN Yun,LIU Hua-lu,LIU Xiu-sheng,SRIBOONCHITTA Songsak.On self-dual constacyclic codes of length ps over Fpm+uFpm[J].Discrete Math,2018,341(2):324-335.
[7]DINH H Q,L′OPEZ-PERMOUTH S R.Cyclic and negacyclic codes over finite chain rings[J].IEEE Trans Inform Theory,2004,50(8):1728-1744.
[8]DOUGHERTY S T,HARADA M,SOL′E P.Self-dual codes over rings and the Chinese remainder theorem[J].Hokkaido Math J,1999,28:253-283.
[9]DOUGHERTY Steven T,KIM Jon-Lark,KULOSMAN Hamid,LIU Hong-wei.Self-dualcodes over commutative Frobenius rings[J].Finite Fields Appl,2010,16:14-26.
[10]DOUGHERTY S T,KIM J-L,LIU H.Constructions of self-dual codes over finite commutative chain rings[J].Int J Inform Coding Theory,2010,1(2):171-190.
[11]DOUGHERTY Steven T,KAYA Abidin,SALT¨URK Esengl.Constructions of self-dualcodes and formally self-dual codes over rings[J].Appl Algebra Engrg Comm Comput,2016,27(5):435-449.
[12]DOUGHERTY Steven T,LEROY Andr,Euclidean self-dual codes over non-commutative Frobenius rings[J].Appl Algebra Engrg Comm Comput,2016,27(3):185-203.
[13]GILDEA Joe,KAYA Abidin,TAYLOR Rhian,YILDIZ Bahattin.Constructions for self-dual codes induced from group rings[J].Finite Fields Appl,2018,51:71-92.
[14]HAN Sunghyu.A method for constructing self-dual codes over Z2m[J].Des Codes Cryptogr,2015,75(2):253-262.
[15]HAMMONS A R,JR.,KUMAR P V,CALDERBANK A R,SLOANE N J A,SOL′E P.The Z4linearity of Kerdock,Preparata,Goethals and related codes[J].IEEE Trans Inform Theory,1994,40(2):301-319.
[16]JITMAN Somphong,LING San,SANGWISUT Ekkasit.On self-dual cyclic codes of length paover GR(p2,s)[J].Adv Math Commun,2016,10(2):255-273.
[17]KARADENIZ Suat,DOUGHERTY Steven T,YILDIZ Bahattin.Constructing formally self-dual codes over Rk[J].Discrete Appl Math,2014,167:188-196.
[18]KIM Boran,LEE Yoonjin.Lee weights of cyclic self-dual codes over Galois rings of characteristic p2[J].Finite Fields Appl,2017,45:107-130.
[19]KIM J-L,LEE Y.Euclidean and Hermitian self-dual MDS codes over large finite fields[J].J Combin Theory,Ser A,2004,105:79-95.
[20]KIM J-L,LEE Y.Construction of MDS self-dual codes over Galois rings[J].Des Codes Cryptogr,2007,45(2):247-258.
[21]KAYA Abidin,YILDIZ Bahattin.Various constructions for self-dual codes over rings and new binary self-dual codes[J].Discrete Math,2016,339(2):460-469.
[22]KAI Xiao-shan,ZHU Shi-xin.Negacyclic self-dual codes over finite chain rings[J].Des Codes Cryptogr,2012,62(2):161-174.
[23]KAI Xiao-shan,ZHU Shi-xin,TANG Yong-sheng.Some constacyclic self-dual codesover the integers modulo2m[J].Finite Fields Appl,2012,18(2):258-270.
[24]LING San,BLACKFORD J.Zpk+1-linear codes[J].IEEE Trans Inform Theory,2002,48:2592-2605.
[25]LIU Xiu-sheng,LIU Hua-lu.Cyclic codes over F2+uF2+vF2[J].Chin Quart J of Math,2014,29(2):189-194.
[26]LIU Xiu-sheng,LIU Hua-lu.The self-dual codes over formal power series rings[J].Chin Quart J of Math,2014,29(1):30-38.
[27]NAGATA Kiyoshi,NEMENZO Fidel,WADA Hideo.Mass formula and structure of self-dual codes over Z2s[J].Des Codes Cryptogr,2013,67(3):293-316.
[28]RAINS E,SLOANE N J A.Self-dual codes,in the Handbook of Coding Theory,V.S.Pless and W.C[J].Huffman,eds.,Elsevier,Amsterdam,1998,177-294.
[29]SHARMA Anuradha,SHARMA Amit K.Construction of self-dual codes over Z2m[J].Cryptogr Commun,2016,8(1):83-101.
[30]WOLFMANN J.Binary image of cyclic codes over Z4[J].IEEE Trans Inform Theory,2001,47(5):1773-1779.
[31]WOOD J.Duality for modules over finite rings and applications to coding theory[J].Amer J Math,1999,121:555-575.
[32]WU Yan-sheng,YUE Qin,LI Feng-wei.Primitive idempotents of irreducible cyclic codes and self-dual cyclic codes over Galois rings[J].Discrete Math,2018,341(6):1755-1767.
[33]YOO Jinjoo,LEE Yoonjin,KIM Boreum.Constructions of formally self-dual codesover Z4and their weight enumerators[J].IEEE Trans Inform Theory,2017,63(12):7667-7675.
[34]ZHANG Guang-hui.Constacyclic codes over Fp+vFp[J].Ars Comb,2018,139:55-67.
[35]ZHANG Guang-hui.A note on self-dual cyclic codes over finite chain rings[J].Adv Math(China),2016,45(2):206-210.
[36]ZHANG Guang-hui,LI Liang-chen.Self-dual codes and cyclic codes over Fp+vFp[J].Ars Comb,2014,116:445-455.
[37]ZHANG Guang-hui,LIU Hong-wei.Self-dual permutation codes over formal power series rings and finite principal ideal rings[J].Acta Math Sci Ser B Engl Ed,2013,33(6):1695-1710.
[38]ZHANG Guang-hui,LIU Qing-qing.Cyclic codes with complementary duals over Fp+vFp[J].Chin Quart J of Math,2016,31(2):118-124.
[39]ZHU Shi-xin,QIAN Kai-yan,KAI Xiao-shan.A note on negacyclic self-dual codesover Z2a[J].Discrete Math,2012,312(22):3270-3275.
[40]ZHU Shi-xin,WANG Li-qi.A class of constacyclic codes over Fp+vFp and its Gray image[J].Discrete Math,2011,311:2677-2682.
[41]ZHU Shi-xin,WANG Yu,SHI Min-jia.Some results on cyclic codes over F2+vF2[J].IEEE Trans Inform Theory,2010,56(4):1680-1684.