形式幂级数环上的自对偶码
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摘要
有限环上纠错码的研究始于上世纪70年代,Blake和Speigel等学者首先研究了整数剩余类环Z_m上的码。接着,Calderbank和Sloane研究了p-进制整数环上的码,进一步,Doughertyl Liu和Park定义了一类类似Z_(p~e)的环R_i同时对这类环和形式幂级数环上的码进行了研究。
本文继续对Dougherty,Liu和Park定义的这类环R_i和形式幂级数环上的码展开了研究,得到了以下主要结果:
在第三章,定义了R_i上的Ⅱ型码,给出了这些码的一些性质和R_i上的自对偶码提升成R_(i+1)上的自对偶码的一个必要条件。
在第四章,我们给出了形式幂级数环上的自对偶码的一种构造方法,并且给出了形式幂级数环上的自对偶码存在的充分必要条件。
The research of error-correcting codes over the finite rings began in 1970s.Blake and Speigel and other researchers had discussed codes over the rings Z_m.Later Calderbank and Sloane discussed codes over the rings p-adic.Further,Dougherty, Liu and Park defined a class of ring,R_i which was similar with the rings Z_(p~e) and the ring of formal power series.Here,R_i= {a_0 + a_1r + a_2r~2 +…+ a_(i-1)r~(i-1)| a_s∈F,(?) 0≤s≤i - 1}.In this definition,F was a finite field and r~(i-1) was equivalent to zero,but r~i was not equivalent to zero.At the same time,the researchers discussed codes over these two class of rings.
In this thesis,we shall continue the stduy on codes over the rings R_i and the ring of formal power series.We obtain the following results.
In chapter 3,we define a typeⅡcode over the rings R_i.We give some properties of this code and a necessary condition for the self-dual codes over R_i to lift to the self-dual codes over R_(i+1).
In chapter 4,we give a method of constructing the self-dual codes over the ring of formal power series,and the sufficient and necessary condition for the exist of the self-dual codes over the ring of formal power series.
本文继续对Dougherty,Liu和Park定义的这类环R_i和形式幂级数环上的码展开了研究,得到了以下主要结果:
在第三章,定义了R_i上的Ⅱ型码,给出了这些码的一些性质和R_i上的自对偶码提升成R_(i+1)上的自对偶码的一个必要条件。
在第四章,我们给出了形式幂级数环上的自对偶码的一种构造方法,并且给出了形式幂级数环上的自对偶码存在的充分必要条件。
The research of error-correcting codes over the finite rings began in 1970s.Blake and Speigel and other researchers had discussed codes over the rings Z_m.Later Calderbank and Sloane discussed codes over the rings p-adic.Further,Dougherty, Liu and Park defined a class of ring,R_i which was similar with the rings Z_(p~e) and the ring of formal power series.Here,R_i= {a_0 + a_1r + a_2r~2 +…+ a_(i-1)r~(i-1)| a_s∈F,(?) 0≤s≤i - 1}.In this definition,F was a finite field and r~(i-1) was equivalent to zero,but r~i was not equivalent to zero.At the same time,the researchers discussed codes over these two class of rings.
In this thesis,we shall continue the stduy on codes over the rings R_i and the ring of formal power series.We obtain the following results.
In chapter 3,we define a typeⅡcode over the rings R_i.We give some properties of this code and a necessary condition for the self-dual codes over R_i to lift to the self-dual codes over R_(i+1).
In chapter 4,we give a method of constructing the self-dual codes over the ring of formal power series,and the sufficient and necessary condition for the exist of the self-dual codes over the ring of formal power series.
引文
[1]Blake.I.F.,Codes over integer residue ring,Inform Contr.,29(1975),295-300.
[2]Calderbank A.R.,Sloane N.J.A.,Modular and p-adic cyclic codes,Desings,Codes,Cryptogr.6(1995),21-35.
[3].Dougherty S.T.,Kim S.Y.,Park Y.H.,Lifted codes and their weight enumerators,Discrete Mathematics,305,Vol 1-3,2005,123-135.
[4]Dougherty S.T.,Liu H.,Cyclic Codes over Formal Power Series Rings,to submitted.
[5]Dougherty S.T.,Liu H.,Independence of Vectors in Codes over Rings,Desings,Codes,Cryptogr.51(2009),55-68.
[6]Dougherty S.T.,Liu H.,Type Ⅱ codes over finite rings,to accepted.
[7]Dougherty S.T.,Liu H.,Park Y.H.,lifts codes over finite chain rings,to preparation.
[8]Dougherty S.T.,Park Y H.,Codes over the p-adic integers,in preparation,2004.
[9]Dougherty S.T.,Shiromoto K.,MDR Codes over Z_k,IEEE Transactions on Information Theory,Volume 46,Number 1,2000,265-269.
[10]Duursma I.M.,Greferath M.,Computing Symmetrized Weight Enumerators for Lifted Quadratic Residue Codes,preprint
[11]Forney G.D.,Sloane N.J.A.,The Nordstrom-Robinson code si a binary Image of the Octacode,DIMACS,14(1993),19-26.
[12]Greferath M.,Schmidt S.E.,Finite-ring combinatorics and Macwilliams'equvialence theorem,J.Combin.Theory,Ser.A,Vol.92,17-28,2000.
[13]Hammons A.R.,Jr.,Kumar P.V.,Calderbank A.R.,Sloane N.J.A.,Sol(?) P.,The Z_4 linearity of Kerdock,Preparata.Goethals and related codes,IEEE-IT,Vol.40,301-319,1994.
[14]Hammons A.R.,Kummar P.V.,Calderbank A.R.,et al.,The Z_4-linearityof Kerdock,Preparata,Goehals and Related code.s,IEEE Trans.Inform.Theory,40(4)(1994),301-319.
[15]Hungerford T.W.,Algebra,Spring-Verlag,Nero York,1974.
[16]Kim J.- L.,Lee Y.,Construction of MDS self-dual codes over Galois Rings,Designs,Codes and Crypto.,Vol.45,247-258,2007.
[17]Kim S.Y.,Liftings of the ternary G01ay code,Master' s Thesis,Kanwon National University,2004.
[18]Mac Williams F.J.,Sloane N.J.A.,The Theory of Error-Correcting Codes.Northholland,Amstterdam,1977.
[19]McDonald B.R.,Finite Rings with Identity,Marcel Dekker,Inc.:New York,1974.
[20]Norton G.H.,S(?)l(?)gean A.,On the Hamming distance of linear codes over a finite chain ring,IEEE-IT,46(3)(2000) 1060-1067.
[21]Pless V.S.,Huffman W.C.,eds.,Handbook of Coding Theory,Elsevier,Amsterdam:1998.
[22]Rains E.,Sloane N.J.A.,Self-dual codes,in the Handbook of Coding Theory,V.S.Pless and W.C.Huffman,eds.:Elsevier:Amsterdam,1998:177-294.
[23]Speigel E.,Codes over Z_m,Inform.Contr.,35(1977),48-52.
[24]Wood J.,Duality for modules over finite rings and applications to coding theory,Amer.J.Math.,Vol.121,No.3,1999,555-575.
[2]Calderbank A.R.,Sloane N.J.A.,Modular and p-adic cyclic codes,Desings,Codes,Cryptogr.6(1995),21-35.
[3].Dougherty S.T.,Kim S.Y.,Park Y.H.,Lifted codes and their weight enumerators,Discrete Mathematics,305,Vol 1-3,2005,123-135.
[4]Dougherty S.T.,Liu H.,Cyclic Codes over Formal Power Series Rings,to submitted.
[5]Dougherty S.T.,Liu H.,Independence of Vectors in Codes over Rings,Desings,Codes,Cryptogr.51(2009),55-68.
[6]Dougherty S.T.,Liu H.,Type Ⅱ codes over finite rings,to accepted.
[7]Dougherty S.T.,Liu H.,Park Y.H.,lifts codes over finite chain rings,to preparation.
[8]Dougherty S.T.,Park Y H.,Codes over the p-adic integers,in preparation,2004.
[9]Dougherty S.T.,Shiromoto K.,MDR Codes over Z_k,IEEE Transactions on Information Theory,Volume 46,Number 1,2000,265-269.
[10]Duursma I.M.,Greferath M.,Computing Symmetrized Weight Enumerators for Lifted Quadratic Residue Codes,preprint
[11]Forney G.D.,Sloane N.J.A.,The Nordstrom-Robinson code si a binary Image of the Octacode,DIMACS,14(1993),19-26.
[12]Greferath M.,Schmidt S.E.,Finite-ring combinatorics and Macwilliams'equvialence theorem,J.Combin.Theory,Ser.A,Vol.92,17-28,2000.
[13]Hammons A.R.,Jr.,Kumar P.V.,Calderbank A.R.,Sloane N.J.A.,Sol(?) P.,The Z_4 linearity of Kerdock,Preparata.Goethals and related codes,IEEE-IT,Vol.40,301-319,1994.
[14]Hammons A.R.,Kummar P.V.,Calderbank A.R.,et al.,The Z_4-linearityof Kerdock,Preparata,Goehals and Related code.s,IEEE Trans.Inform.Theory,40(4)(1994),301-319.
[15]Hungerford T.W.,Algebra,Spring-Verlag,Nero York,1974.
[16]Kim J.- L.,Lee Y.,Construction of MDS self-dual codes over Galois Rings,Designs,Codes and Crypto.,Vol.45,247-258,2007.
[17]Kim S.Y.,Liftings of the ternary G01ay code,Master' s Thesis,Kanwon National University,2004.
[18]Mac Williams F.J.,Sloane N.J.A.,The Theory of Error-Correcting Codes.Northholland,Amstterdam,1977.
[19]McDonald B.R.,Finite Rings with Identity,Marcel Dekker,Inc.:New York,1974.
[20]Norton G.H.,S(?)l(?)gean A.,On the Hamming distance of linear codes over a finite chain ring,IEEE-IT,46(3)(2000) 1060-1067.
[21]Pless V.S.,Huffman W.C.,eds.,Handbook of Coding Theory,Elsevier,Amsterdam:1998.
[22]Rains E.,Sloane N.J.A.,Self-dual codes,in the Handbook of Coding Theory,V.S.Pless and W.C.Huffman,eds.:Elsevier:Amsterdam,1998:177-294.
[23]Speigel E.,Codes over Z_m,Inform.Contr.,35(1977),48-52.
[24]Wood J.,Duality for modules over finite rings and applications to coding theory,Amer.J.Math.,Vol.121,No.3,1999,555-575.