自正交矩阵乘积线性码
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摘要
基于矩阵乘积结构构造自正交码,给出了矩阵乘积线性码是自正交码的一个必要条件.指出了在输入码是嵌套结构时,自正交矩阵乘积线性码的基本矩阵与其转置矩阵的乘积不必是对角矩阵,并给出了一些例子.此外,还研究了自对偶矩阵乘积线性码.
The matrix-product construction is applied to construct self-orthogonal codes. A necessary condition for constructing self-orthogonal matrix-product codes is given. It is pointed out that under the condition that the input codes are nested, the product of the underlying matrix and its transposed matrix need not be a diagonal matrix. Some illustrative examples are given. In addition, self-dual matrix-product codes are also studied.
The matrix-product construction is applied to construct self-orthogonal codes. A necessary condition for constructing self-orthogonal matrix-product codes is given. It is pointed out that under the condition that the input codes are nested, the product of the underlying matrix and its transposed matrix need not be a diagonal matrix. Some illustrative examples are given. In addition, self-dual matrix-product codes are also studied.
引文
[1] Jin L,Ling S,Lou J,etal.Application of classical Hermitian self-orthogonal MDS codes to quantum MDS codes[J].IEEE Trans.Inf.Theory,2010,53(9):4735-4740 .
[2] Jin L,Xing C.Euclidean and Hermitian self-orthogonal algebraic geometry and their application to quantum codes[J].IEEE Trans.Inf.Theory,2012,58:5484-5489.
[3] Pless V.A classification of self-orthogonal codes over GF(2)[J].Discrete Math,1972,3:209-246.
[4] 晁海洲,倪翔飞.基于自对偶码的LDPC码构造(英文)[J].大学数学,2016,32(1):7-10.
[5] 王永.环F2+uF2+…+ukF2上的Type Ⅱ码[J].大学数学,2015,31(3):97-101.
[6] Blackmore T,Norton G H.Matrix-product code over Fq [J].Applicable Algebra in Engineering,Communication and Computing,2001,12(6):477-500.
[7] Bouye M,Sa D M.Matrix-product codes over Fp [J].Gulf J.math.,2015,3(2):44-48.
[8] Galindo C,Hemando F,Ruano D.New quantum codes from evaluation and matrix-product codes[J].Finite Fields Appl.,2015,36:98-120.
[9] Liu X,Dinh H Q,Liu H,etal.On new quantum codes from matrix-product codes[J].Cryptography and Communications,2018,10(4):579-589.
[10] Mankean T,Jitman S.Matrix-product constructions for self-orthogonal linear codes[C]// in IEEE Conference Publications,2016 12th International Conference on Mathematics,Statistics,and Their Applications (ICMSA),Banda Aceh,Indonesia,2016:6-10.
[11] Hernando F,Lally K,Ruano D.Construction and decoding of matrix-product codes from nested codes[J].Applicable Algebra in Engineering,Communication and Computing,2009,20(5/6):497-507.
[12] Ezerman M F,Jitman S,Kiah H M,etal.Pure asymmetric quantum MDS codes from CSS construction:A complete characterization[J].Int.J.of Quantum Information,2013,11(3):1350027(1-10).
[13] Hernando F,Ruano D.New linear codes from matrix-product codes with polynomial units[J].Advancess in Mathematics of Communications,2010,4(3):363-367.
[14] Mddeni M B O,Souidi E M.Construction and bound on performance of matrix-product codes[J].Applied Mathematical Sciences,2011,5(19):929-934.
[2] Jin L,Xing C.Euclidean and Hermitian self-orthogonal algebraic geometry and their application to quantum codes[J].IEEE Trans.Inf.Theory,2012,58:5484-5489.
[3] Pless V.A classification of self-orthogonal codes over GF(2)[J].Discrete Math,1972,3:209-246.
[4] 晁海洲,倪翔飞.基于自对偶码的LDPC码构造(英文)[J].大学数学,2016,32(1):7-10.
[5] 王永.环F2+uF2+…+ukF2上的Type Ⅱ码[J].大学数学,2015,31(3):97-101.
[6] Blackmore T,Norton G H.Matrix-product code over Fq [J].Applicable Algebra in Engineering,Communication and Computing,2001,12(6):477-500.
[7] Bouye M,Sa D M.Matrix-product codes over Fp [J].Gulf J.math.,2015,3(2):44-48.
[8] Galindo C,Hemando F,Ruano D.New quantum codes from evaluation and matrix-product codes[J].Finite Fields Appl.,2015,36:98-120.
[9] Liu X,Dinh H Q,Liu H,etal.On new quantum codes from matrix-product codes[J].Cryptography and Communications,2018,10(4):579-589.
[10] Mankean T,Jitman S.Matrix-product constructions for self-orthogonal linear codes[C]// in IEEE Conference Publications,2016 12th International Conference on Mathematics,Statistics,and Their Applications (ICMSA),Banda Aceh,Indonesia,2016:6-10.
[11] Hernando F,Lally K,Ruano D.Construction and decoding of matrix-product codes from nested codes[J].Applicable Algebra in Engineering,Communication and Computing,2009,20(5/6):497-507.
[12] Ezerman M F,Jitman S,Kiah H M,etal.Pure asymmetric quantum MDS codes from CSS construction:A complete characterization[J].Int.J.of Quantum Information,2013,11(3):1350027(1-10).
[13] Hernando F,Ruano D.New linear codes from matrix-product codes with polynomial units[J].Advancess in Mathematics of Communications,2010,4(3):363-367.
[14] Mddeni M B O,Souidi E M.Construction and bound on performance of matrix-product codes[J].Applied Mathematical Sciences,2011,5(19):929-934.