F_(4~m)上厄米特自正交常循环码
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摘要
有限域上常循环码具有丰富的代数结构,其编译码电路容易实现,因而在信息传输实践中具有重要的应用.该文研究了一类有限域上任意长度的厄米特自正交常循环码的结构,给出了此类有限域上厄米特自正交常循环码的生成多项式与存在条件,确立了此类有限域上厄米特自正交常循环码的计数公式,并且利用此类有限域上偶长度的厄米特自正交常循环码构造了最优的量子码.
Constacyclic codes over finite fields are a class of important linear codes. This class of codes has rich algebra structure and its encoding and decoding circuits can be easily performed. Constacyclic codes over finite fields have many applications in information transmission. In this paper,the structure of Hermitian self-orthogonal constacyclic codes over a class of finite fields of any length is studied. By using generator polynomial,the condition for the existence of Hermitian self-orthogonal constacyclic codes over this class of finite fields is explored and the enumeration formula of such codes is determined. Further,Hermitian self-orthogonal constacyclic codes over this class finite fields are applied to construct some optimal quantum codes.
Constacyclic codes over finite fields are a class of important linear codes. This class of codes has rich algebra structure and its encoding and decoding circuits can be easily performed. Constacyclic codes over finite fields have many applications in information transmission. In this paper,the structure of Hermitian self-orthogonal constacyclic codes over a class of finite fields of any length is studied. By using generator polynomial,the condition for the existence of Hermitian self-orthogonal constacyclic codes over this class of finite fields is explored and the enumeration formula of such codes is determined. Further,Hermitian self-orthogonal constacyclic codes over this class finite fields are applied to construct some optimal quantum codes.
引文
[1]Jia Y,Ling S,Xing C.On self-dual cyclic codes over finite fields[J].IEEE Transactions on Information Theory,2011,57(4):2243-2251.
[2]Yang Y,Cai W.On self-dual constacyclic codes over finite fields[J].Designs,Codes and Cryptography,2015,74(2):355-365.
[3]Shi M,Zhang Y.Quasi-twisted codes with constacyclic constituent codes[J].Finite Fields and Their Application,2016,39(3):159-178.
[4]Chao Y.On constacyclic codes over finite chain rings[J].Finite Fields and Their Application,2013,24(24):124-135.
[5]施敏加.环F2+u F2+…+uk-1F2上常循环自对偶码[J].电子学报,2013,41(6):1088-1092.Shi M.Constacyclic self-dual codes over ring F2+u F2+…+uk-1F2[J].Acta Electronica Sinica,2013,41(6):1088-1092.(in Chinese)
[6]施敏加,杨善林,朱士信.环F2+u F2上长度为2e的循环码的距离[J].电子学报,2011,39(1):29-34.Shi M,Yang S,Zhu S.On minimum distances of cyclic codes of length 2eover F2+u F2[J].Acta Electronica Sinica,2011,39(1):29-34.(in Chinese)
[7]Shi M,Zhu S,Yang S.A class of optimal p-ary codes from one-weight codes over Fp[u]/〈um〉[J].Journal of the Franklin Institute,2013,350(5):729-737.
[8]Chen B,Ling S,Zhang G.Applications of constacyclic codes of quantum MDS codes[J].IEEE Transactions on Information Theory,2015,61(3):1474-1484.
[9]Kai X,Zhu S,Li P.Constacyclic codes and some new MDS quantum codes[J].IEEE Transactions on Information Theory,2014,60(4):2080-2085.
[10]Sloane N J A,Thompson J G.Cyclic self-dual codes[J].IEEE Transactions on Information Theory,1983,5(3):364-366.
[11]Kai X,Zhu S.On cyclic self-dual codes[J].Applicable Algebra in Engineering,Communication and Computing,2008,19(6):509-525.
[12]Bakshi G K,Raka M.Self-dual and self-orthogonal negacyclic codes of length 2pnover a finite field[J].Finite Fields and Their Applications,2013,19(1):39-54.
[13]张付丽,开晓山,朱士信,陈安顺.一种有限域上自正交码的构造方法[J].电子与信息学报,2014,36(10):2326-2330.Zhang F,Kai X,Zhu S,Chen A.A method for constructing self-orthogonal codes over finite fields[J].Journal of Electronics&Information Technology,2014,36(10):2326-2330.(in Chinese)
[14]Calderbank A R,Rains E M,Shor P W,Sloane N J A.Quantum error correction via codes over[J].IEEE Transactions on Information Theory,1998,44(4):1369-1387.
[15]Ashikhmin A,Knill E.Nonbinary quantum stablizer codes[J].IEEE Transactions on Information Theory,2001,47(7):3065-3072.
[16]Blackford T.Negacyclic duadic codes[J].Finite Fields and Their Applications,2008,14(4):930-943.
[17]Sahni A,Sehgal P T.Hermitian self-orthogonal constacyclic codes over finite fields[J].Journal of Discrete Mathematics,2014,2014(1):1-7.
[18]李卓,邢莉娟,王新梅.量子常数循环码[J].西安电子科技大学学报(自然科学版),2009,36(1):48-51.Li Z,Xing L,Wang X.Quantum constacyclic codes[J].Journal of Xidian University,2009,36(1):48-51.(in Chinese)
[19]李瑞虎,左飞,刘杨.斜对称q2-分圆陪集及其应用研究[J].空军工程大学学报(自然科学版),2011,12(1):87-89.Li R,Zuo F,Liu Y.A study of skew symmetric q2-cyclotomic coset and its application[J].Journal of Air Force Engineering University(Natural Science Edition),2011,12(1):87-89.(in Chinese)
[20]钱建发,马文平.量子纠错码的一个统一构造方法[J].计算机科学,2010,37(3):70-72.Qian J,Ma W.Unified approach to construct quantum errorcorrecting code[J].Computer Science,2010,37(3):70-72.(in Chinese)
[21]Jin L,Xing C.A construction of new quantum MDS codes[J].IEEE Transactions on Information Theory,2014,60(5):2921-2925.
[22]Grassl M.Bounds on the Minimum Distance of Linear Codes and Quantum Codes[OL].http://www.codetables.de,2015,12.
[2]Yang Y,Cai W.On self-dual constacyclic codes over finite fields[J].Designs,Codes and Cryptography,2015,74(2):355-365.
[3]Shi M,Zhang Y.Quasi-twisted codes with constacyclic constituent codes[J].Finite Fields and Their Application,2016,39(3):159-178.
[4]Chao Y.On constacyclic codes over finite chain rings[J].Finite Fields and Their Application,2013,24(24):124-135.
[5]施敏加.环F2+u F2+…+uk-1F2上常循环自对偶码[J].电子学报,2013,41(6):1088-1092.Shi M.Constacyclic self-dual codes over ring F2+u F2+…+uk-1F2[J].Acta Electronica Sinica,2013,41(6):1088-1092.(in Chinese)
[6]施敏加,杨善林,朱士信.环F2+u F2上长度为2e的循环码的距离[J].电子学报,2011,39(1):29-34.Shi M,Yang S,Zhu S.On minimum distances of cyclic codes of length 2eover F2+u F2[J].Acta Electronica Sinica,2011,39(1):29-34.(in Chinese)
[7]Shi M,Zhu S,Yang S.A class of optimal p-ary codes from one-weight codes over Fp[u]/〈um〉[J].Journal of the Franklin Institute,2013,350(5):729-737.
[8]Chen B,Ling S,Zhang G.Applications of constacyclic codes of quantum MDS codes[J].IEEE Transactions on Information Theory,2015,61(3):1474-1484.
[9]Kai X,Zhu S,Li P.Constacyclic codes and some new MDS quantum codes[J].IEEE Transactions on Information Theory,2014,60(4):2080-2085.
[10]Sloane N J A,Thompson J G.Cyclic self-dual codes[J].IEEE Transactions on Information Theory,1983,5(3):364-366.
[11]Kai X,Zhu S.On cyclic self-dual codes[J].Applicable Algebra in Engineering,Communication and Computing,2008,19(6):509-525.
[12]Bakshi G K,Raka M.Self-dual and self-orthogonal negacyclic codes of length 2pnover a finite field[J].Finite Fields and Their Applications,2013,19(1):39-54.
[13]张付丽,开晓山,朱士信,陈安顺.一种有限域上自正交码的构造方法[J].电子与信息学报,2014,36(10):2326-2330.Zhang F,Kai X,Zhu S,Chen A.A method for constructing self-orthogonal codes over finite fields[J].Journal of Electronics&Information Technology,2014,36(10):2326-2330.(in Chinese)
[14]Calderbank A R,Rains E M,Shor P W,Sloane N J A.Quantum error correction via codes over[J].IEEE Transactions on Information Theory,1998,44(4):1369-1387.
[15]Ashikhmin A,Knill E.Nonbinary quantum stablizer codes[J].IEEE Transactions on Information Theory,2001,47(7):3065-3072.
[16]Blackford T.Negacyclic duadic codes[J].Finite Fields and Their Applications,2008,14(4):930-943.
[17]Sahni A,Sehgal P T.Hermitian self-orthogonal constacyclic codes over finite fields[J].Journal of Discrete Mathematics,2014,2014(1):1-7.
[18]李卓,邢莉娟,王新梅.量子常数循环码[J].西安电子科技大学学报(自然科学版),2009,36(1):48-51.Li Z,Xing L,Wang X.Quantum constacyclic codes[J].Journal of Xidian University,2009,36(1):48-51.(in Chinese)
[19]李瑞虎,左飞,刘杨.斜对称q2-分圆陪集及其应用研究[J].空军工程大学学报(自然科学版),2011,12(1):87-89.Li R,Zuo F,Liu Y.A study of skew symmetric q2-cyclotomic coset and its application[J].Journal of Air Force Engineering University(Natural Science Edition),2011,12(1):87-89.(in Chinese)
[20]钱建发,马文平.量子纠错码的一个统一构造方法[J].计算机科学,2010,37(3):70-72.Qian J,Ma W.Unified approach to construct quantum errorcorrecting code[J].Computer Science,2010,37(3):70-72.(in Chinese)
[21]Jin L,Xing C.A construction of new quantum MDS codes[J].IEEE Transactions on Information Theory,2014,60(5):2921-2925.
[22]Grassl M.Bounds on the Minimum Distance of Linear Codes and Quantum Codes[OL].http://www.codetables.de,2015,12.