一类新的q元量子MDS码
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摘要
量子纠错码在量子信息处理和量子计算中有着重要的应用.q元量子MDS码是一类重要的最优量子纠错码,此类量子码的参数满足相应的量子Singleton界.构造q元量子MDS码具有重要的理论和应用意义.但构造码长q+1的q元量子MDS码是比较困难的,许多码长(q+1)(q-1)/m的q元量子MDS码,其中m整除q+1或q-1,已经被构造出来.在HE Xiangming等构造出的q元量子MDS码的基础上,给出了几类q元量子MDS码的具体实例,这些量子MDS码具有码长(q+1)(q-1)/m,其中m整除(q+1)(q-1),但m不整除q-1,也不整除q+1.
Quantum codes have applications in quantum computing and quantum communications.Quantum maximal distance separable(MDS)codes are a class of optimal quantum error-correcting codes and their parameters satisfy the quantum Singleton bound.The construction of quantum MDS codes has important application in theory and practice.It is still difficult to construct q-ary quantum MDS codes of length bigger than q+1with a big minimum distance.Many q-ary quantum MDS codes of length(q+1)(q-1)/m have been constructed,where mis a factor of q+1 or q-1.This paper uses some results in[20]and presents some quantum MDS codes of length(q+1)(q-1)/m,where m is a factor of(q+1)(q-1),m is not a factor of q+1or q-1.
Quantum codes have applications in quantum computing and quantum communications.Quantum maximal distance separable(MDS)codes are a class of optimal quantum error-correcting codes and their parameters satisfy the quantum Singleton bound.The construction of quantum MDS codes has important application in theory and practice.It is still difficult to construct q-ary quantum MDS codes of length bigger than q+1with a big minimum distance.Many q-ary quantum MDS codes of length(q+1)(q-1)/m have been constructed,where mis a factor of q+1 or q-1.This paper uses some results in[20]and presents some quantum MDS codes of length(q+1)(q-1)/m,where m is a factor of(q+1)(q-1),m is not a factor of q+1or q-1.
引文
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[16]KAI X,ZHU S.New quantum MDS codes from negacyclic codes[J].Information Theory,IEEE Transactions on,2013,59(2):1193-1197.
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[18]CHEN B,LING S,ZHANG G.Application of constacyclic codes to quantum MDS codes[J].Information Theory,IEEE Transactions on,2015,61(3):1474-1484.
[19]KAI X,ZHU S,LI P.Constacyclic codes and some new quantum MDS codes[J].Information Theory,IEEE Transactions on,2014,60(4):2080-2086.
[20]HE X,XU L,CHEN H.New q-ary Quantum MDS Codes with Distances Bigger than q/2[J].arXiv preprint arXiv:1507.08355,2015.
[2]ASHIKHMIN A,KNILL E.Nonbinary quantum stabilizer codes[J].Information Theory,IEEE Transactions on,2001,47(7):3065-3072.
[3]ASHIKHMIN A,LITSYN S.Foundations of quantum error correction[J].AMS IP STUDIES IN ADVANCED MATHEMATICS,2007,41:151-185.
[4]ASHIKHMIN A,TSFASMAN M A,LITSYN S.Asymptotically good quantum codes[J].Physical Review A,2000,63(3):222-224.
[5]BIERBRAUER J,EDEL Y.Quantum twisted codes[J].Journal of Combinatorial Designs,2000,8(3):174-188.
[6]FENG K,LING S,XING C.Asymptotic bounds on quantum codes from algebraic geometry codes[J].Information Theory,IEEE Transactions on,2006,52(3):986-991.
[7]GRASSL M,BETH T,ROETTELER M.On optimal quantum codes[J].International Journal of Quantum Information,2004,2(1):55-64.
[8]LA GUARDIA G G.Constructions of new families of nonbinary quantum codes[J].Physical Review A,2009,80(4):3383-3387.
[9]LA GUARDIA G G.New quantum MDS codes[J].Information Theory,IEEE Transactions on,2011,57(8):5551-5554.
[10]LA GUARDIA G G,PALAZZO R.Constructions of new families of nonbinary CSS codes[J].Discrete Mathematics,2010,310(21):2935-2945.
[11]HAMADA M.Concatenated quantum codes constructible in polynomial time:Efficient decoding and error correction[J].Information Theory,IEEE Transactions on,2008,54(12):5689-5704.
[12]CALDERBANK A R,RAINS E M,SHOR P W,et al.Quantum error correction via codes over GF(4)[J].Information Theory,IEEE Transactions on,1996,44(4):1369-1387.
[13]FENG K.Quantum codes[[6,2,3]]p and[[7,3,3]]p(p≥3)exist[J].Information Theory,IEEE Transactions on,2002,48(8):2384-2391.
[14]JIN L,LING S,LUO J,et al.Application of classical Hermitian self-orthogonal MDS codes to quantum MDS codes[J].Information Theory,IEEE Transactions on,2010,56(9):4735-4740.
[15]JIN L,XING C.A construction of new quantum MDS codes[J].Information Theory,IEEE Transactions on,2014,60(5):2921-2925.
[16]KAI X,ZHU S.New quantum MDS codes from negacyclic codes[J].Information Theory,IEEE Transactions on,2013,59(2):1193-1197.
[17]JIN L,XING C.Euclidean and Hermitian self-orthogonal algebraic geometry codes and their application to quantum codes[J].Information Theory,IEEE Transactions on,2012,58(8):5484-5489.
[18]CHEN B,LING S,ZHANG G.Application of constacyclic codes to quantum MDS codes[J].Information Theory,IEEE Transactions on,2015,61(3):1474-1484.
[19]KAI X,ZHU S,LI P.Constacyclic codes and some new quantum MDS codes[J].Information Theory,IEEE Transactions on,2014,60(4):2080-2086.
[20]HE X,XU L,CHEN H.New q-ary Quantum MDS Codes with Distances Bigger than q/2[J].arXiv preprint arXiv:1507.08355,2015.