有限链环上一类常循环码的距离
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摘要
在编码理论中,线性码的(最小)距离是一个极其重要的参数,它决定了码的纠错能力。设R为任一有限交换链环,a为其最大理想的一个生成元,R~*为R的乘法单位群。对于任意w?R~*,该文利用R上任意长度的(1+aw)-常循环码的生成结构,通过计算这类码的高阶挠码,得到了R上任意长度的(1+aw)-常循环码的汉明距离,并研究了这类常循环码的齐次距离。这给编译有限链环上此类常循环码提供了重要的理论依据。
In coding theory, the(minimum) distance of a code is a very important invariant, which always determines the error-correcting capability of the code. Let R be an arbitrary commutative finite chain ring, a is a generator of the unique maximal ideal and R~* is the multiplicative group of units of R. In this paper, for any w?R~*, by using the generator polynomials of(1 +aw)-constacyclic codes of any length over R, higher torsion codes of such codes are calculated. The Hamming distance of all(1 +aw)-constacyclic codes of any length over R is determined and the exact homogeneous distance of some such codes is obtained. The result provides a theoretical basis for encoding and decoding for such constacyclic codes.
In coding theory, the(minimum) distance of a code is a very important invariant, which always determines the error-correcting capability of the code. Let R be an arbitrary commutative finite chain ring, a is a generator of the unique maximal ideal and R~* is the multiplicative group of units of R. In this paper, for any w?R~*, by using the generator polynomials of(1 +aw)-constacyclic codes of any length over R, higher torsion codes of such codes are calculated. The Hamming distance of all(1 +aw)-constacyclic codes of any length over R is determined and the exact homogeneous distance of some such codes is obtained. The result provides a theoretical basis for encoding and decoding for such constacyclic codes.
引文
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[5]DINH H Q,DHOMPONGSA S,and SRIBOONCHITTA S.Repeated-root constacyclic codes of prime power length over[]m a pF u u and their duals[J].Discrete Mathematics,2016,339(6):1706-1715.doi:10.1016/j.disc.2016.01.020.
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[12]DINH H Q.Constacyclic codes of length sp over mpF mp+u F[J].Journal of Algebra,2010,324(5):940-950.doi:10.1016/j.jalgebra.2010.05.027.
[2]施敏加,杨善林,朱士信.环12 2 2kF u F u F-++?+上长度为2s的(1+u)-常循环码的距离分布[J].电子与信息学报,2010,32(1):112-116.doi:10.3724/SP.J.1146.2008.01810.SHI M J,YANG S L,and ZHU S X.The distributions of distances of(1+u)-constacyclic codes of length 2s over12 2 2kF u F u F-++?+[J].Journal of Electronics&Information Technology,2010,32(1):112-116.doi:10.3724/SP.J.1146.2008.01810.
[3]KONG B,ZHENG X Y,and MA H J.The depth spectrums of constacyclic codes over finite chain rings[J].Discrete Mathematics,2015,338(2):256-261.doi:10.1016/j.disc.2014.09.013.
[4]QIAN K Y,ZHU S X,and KAI X S.On cyclic self-orthogonal codes over 2mZ[J].Finite Fields and Their Applications,2015,33:53-65.doi:10.1016/j.ffa.2014.11.005.
[5]DINH H Q,DHOMPONGSA S,and SRIBOONCHITTA S.Repeated-root constacyclic codes of prime power length over[]m a pF u u and their duals[J].Discrete Mathematics,2016,339(6):1706-1715.doi:10.1016/j.disc.2016.01.020.
[6]WOLFMANN J.Negacyclic and cyclic codes over 4Z[J].IEEE Transactions on Information Theory,1999,45(7):2527-2532.doi:10.1109/18.796397.
[7]NORTON G H and v vSALAGAN A.On the structure of linear and cyclic codes over a finite chain ring[J].Applicable Algebra in Engineering,Communication and Computing,2000,10(6):489-506.doi:10.1007/PL00012382.
[8]NORTON G H and v vSALAGAN A.On the Hamming distance of linear codes over a finite chain ring[J].IEEE Transactions on Information Theory,2000,46(3):1060-1067.doi:10.1109/18.841186.
[9]GREFERATH M and SCHMIDT S E.Gray isometries for finite chain rings and a nonlinear ternary 12(36,3,15)code[J].IEEE Transactions on Information Theory,1999,45(7):2522-2524.doi:10.1109/18.796395.
[10]CAO Y L.On constacyclic codes over finite chain rings[J].Finite Fields and Their Applications,2013,24:124-135.doi:10.1016/j.ffa.2013.07.001.
[11]MCDONALD B R.Finite Rings with Identity[M].New York,Marcel Dekker Press,1974:56-97.
[12]DINH H Q.Constacyclic codes of length sp over mpF mp+u F[J].Journal of Algebra,2010,324(5):940-950.doi:10.1016/j.jalgebra.2010.05.027.