环Z_4上自对偶码的构造
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摘要
利用有限环Z_4+v Z_4(其中v_2=1)上自对偶码,给出了一种构造Z_4上自对偶码的方法.引入了(Z_4+v Z_4)n到Z_2n_4的保距Gray映射,给出了Z_4+v Z_4上自对偶码的性质,证明了Z_4+v Z_4上长为n的自对偶码的Gray像是Z_4上长为_2n的自对偶码,由此构造了Z_4上一些极优的类型I与类型Ⅱ自对偶码.
A M ethod for Construction self-dual codes over Z_4 is proposed by using self-dual codes over the ring Z_4+v Z_4,where v_2= 1. A Gray map from( Z_4+ v Z_4)nto Z_2n_4 is introduced,and some properties about self-dual codes over Z_4+v Z_4 are given. It show ns that the Gray image of a self-dual code over Z_4+ v Z_4 of length n is a self-dual code over Z_4 of length _2n. Further,some extremal Type I and Type Ⅱ codes over Z_4 are constructed.
A M ethod for Construction self-dual codes over Z_4 is proposed by using self-dual codes over the ring Z_4+v Z_4,where v_2= 1. A Gray map from( Z_4+ v Z_4)nto Z_2n_4 is introduced,and some properties about self-dual codes over Z_4+v Z_4 are given. It show ns that the Gray image of a self-dual code over Z_4+ v Z_4 of length n is a self-dual code over Z_4 of length _2n. Further,some extremal Type I and Type Ⅱ codes over Z_4 are constructed.
引文
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[9]Yildiz B,Karadeniz S.Linear codes over Z4+u Z4:MacWilliams identities,projections,and formally self-dual codes[J].Finite Fields and Their Applications,2014,27:24-40.
[10]Wood J.Duality for modules over finite rings and applications to coding theory[J].The American Journal of M athematics,1999,121(3):555-575.
[11]Doughterty S T,Kim J L,Kulosman H,Liu H W.Selfdual codes over commutative Frobenius rings[J].Finite Fields and Their Applications,2010,16(1):14-26.
[2]吴波,朱士信,李平.环Fp+u Fp上的Kerdock码与Preparata码[J].电子学报,2008,36(7):1364-1367.Wo Bo,Zhu Shi-xin,Li Ping.Kerdock codes and Preparata codes over rings Fp+u Fp[J].Acta Electronica Sinica,2008,36(7):1364-1367.(in Chinese)
[3]朱士信,许和乾,施敏加.环Z4上线性码关于RT距离M ac Walliams恒等式[J].电子学报,2009,37(5):1115-1118.Zhu Shi-xin,Xu He-qian,Shi M in-jia.M ac Walliams identities of linear codes over ring Z4w ith respect to the RT metric[J].Acta Electronica Sinica,2009,37(5):1115-1118.(in Chinese)
[4]施敏加,杨善林.非主理想环Fp+v Fp上线性码的MacWalliams恒等式[J].电子学报,2011,39(10):2449-2453.Shi M in-jia,Yang Shan-lin.M ac Williams identities of linear codes over non-principal ideal ring Fp+v Fp[J].Acta Electronica Sinica,2011,39(10):2449-2453.(in Chinese)
[5]Bachoc C.Applications of coding theory to the construction of modular lattices[J].Journal of Combinatorial Theory Series A,1997,78(1):92-119.
[6]Bonnecaze A,SoléP,Bachoc C,Mourrain B.TypeⅡcodes over Z4[J].IEEE Transactions on Information Theory,1997,43(3):969-976.
[7]Dougherty S T,Gaborit P,Harada M,SoléP.TypeⅡcodes over F2+u F2[J].IEEE Transactions on Information Theory,1999,45(1):32-45.
[8]Yildiz B,Karadeniz S.Self-dual codes over F2+u F2+v F2+uv F2[J].Journal of the Franklin Institute,2010,347(10):1888-1894.
[9]Yildiz B,Karadeniz S.Linear codes over Z4+u Z4:MacWilliams identities,projections,and formally self-dual codes[J].Finite Fields and Their Applications,2014,27:24-40.
[10]Wood J.Duality for modules over finite rings and applications to coding theory[J].The American Journal of M athematics,1999,121(3):555-575.
[11]Doughterty S T,Kim J L,Kulosman H,Liu H W.Selfdual codes over commutative Frobenius rings[J].Finite Fields and Their Applications,2010,16(1):14-26.