| Quantum error-correction is necessary to protect quantum information from errors due to decoherence and other quantum noise.The design of good quantum error-correction codes(QECC)is crucial for the future quantum computation and quantum communication.In this dissertation,several constructions of QECCs are conducted,and the following research results are obtained(1)Research on the nested/dual containing relationship between several subclasses of clas-sical Alternant codes is carried out.Firstly,a new subclass of binary Alternant codes,which can asymptotically meets the Gilbert-Varshamov bound,is proposed based on the Hamming weight distribution of maximum-distance-separable(MDS)codes.Then,the nested containing relationship between BCH codes and the subclass Alternant codes,and also the relationship between BCH codes and Chien-Choy GBCH codes,are obtained.Thus,several classes of QECCs can be constructed based on the dual containing relation-ship between BCH codes.At last,asymptotically good binary expansions of quantum GRS codes are obtained.(2)Five classes of q-ary entanglement-assisted quantum MDS(EAQMDS)codes with min-imum distance greater than q + 1 are constructed,whose parameters are listed as follows?[[q2 + 1,q2-2d + 4,d;1]]q,where q is a prime power,2 ? d ? 2q is an even integer.?[[q2,q2-2d + 3.d;1]]q,where q is a prime power,q + 1 ? d ? 2q-1.?[[q2-1,q2-2d + 2,d;1]]q,where q is a prime power,2?d? 2q-2.?[[q2-1/2,q2-1/2-2d+4,d;2]]q,where q is an odd prime powerq+1/2+2?d?3/2q-1/2.?[[q2-1/t,q2-1/t-2d+t+2,d;t]]q,where q is an odd prime power with t?(q+1),t?3 is an odd integer,and(t-1)(q+1)/t+2?d?(t+1)(q+1)/t-2.EAQMDS codes in ??? have minimum distance upper limit greater than q + 1 by con-suming a few pre-shared maximally entangled states.These codes have broken through the limit that the minimum distance of q-ary standard quantum MDS codes is less than or equal to q + 1.In particular,each code in ??? has nearly double minimum distance upper limit of the standard QMDS code of the same length constructed so far,and con-sumes only one pair of maximally entangled states.Therefore,these codes are significant in the future quantum communication.(3)A general framework for the construction of quantum tensor product codes(QTPC)is proposed.By adding some constraints on the component codes,several classes of dual containing tensor product codes(TPC)are obtained.Through selecting different types of component codes,the proposed method enables the construction of a large family of QTPCs,and they can provide the similar error-correction,error-detection or error-location abilities as the corresponding classical TPCs.In particular,if one of the component codes is selected as a burst error-correction code,then QTPCs have multiple quantum burst error-correction abilities,provided these bursts fall in distinct subblocks.Compared with concatenated quantum codes(CQC),the component code selections of QTPCs are much more flexible than those of CQCs since only one of the component codes of QTPCs needs to satisfy the dual-containing restriction.Furthermore,we show that it is possible to construct QTPCs with parameters better than other classes of quantum error-correction codes(QECC),e.g.,CQCs and quantum BCH codes.It is known that classical TPCs cannot have parameters better than classical BCH codes.However,we show that QTPCs can have better parameters than quantum BCH codes.Many QTPCs are obtained with parameters better than previously known QECCs available in the literature.(4)Several corrections and amendments are given to the proof of the equivalence of general-ized concatenated codes(GCC codes)and generalized error-location codes(GEL codes),in which the construction of quantum tensor product codes is very close to the equivalence proof. |
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