Construction Of Quantum BCH Codes

When a quantum system is not perfectly isolated,quantum decoherence phenomenon de-stroys quantum states carrying information.Quantum error correcting codes play a crucial role in quantum decoherence.How to construct quantum codes with good parameters is an important part in the fields of quantum information theory.There exist links to classical coding theory that generates quantum codes.Nevertheless,it is known results that choose cyclotomic cosets from given parameters.There are lack of common methods.Furthermore,known conclusions have more restrictions placed on constructions such as finite field with odd prime power and given code length.Consequently,quantum codes can be obtained in the certain length ranges.In order to overcome these shortcomings,a general method of well-chosen cosets is introduced with the support of relevant theories in this thesis.On this basis,the dissertation designs quantum BCH codes with any length over finite field.It fails to operate on any quantum system in virtue of finite fields with prime power order.Fortunately,finite rings are able to work.Recently,there is a little research on quantum error-correcting codes over finite ring.Quantum codes over finite rings are presented in this thesis for sake of correcting errors on any quantum system.This dissertation proposes construction algorithm on finite chain rings.In particular,two schemes are put forward over Galois rings.Furthermore,relationship on quantum codes between finite fields and finite rings is analysed briefly.The main contributions are presented as follows:(1)The dissertation comes up with constructions on quantum BCH codes with prime order.When paying attention to order two over finite fields,quantum codes with length n=r(q-1)have been constructed.The paper studies its mirror length n=r(q+1)complementing known results.Quantum maximum distance separable codes with minimum distance three can only be found over F_q2by Hermitian construction.This dissertation designs quantum maximum distance separable codes with minimum distance 3 applied to Steane's construc-tion over any finite fields.It enriches quantum error correcting codes theory.With regard to order three,known conclusions are extend to more general situation so that constructions have fewer restrictions on cyclotomic cosets.Furthermore,rang of given cyclotomic cosets is much wider.Compared with the ones available in the literature,new quantum BCH codes can be obtained with higher dimensions and better minimum distances lower bound.(2)Quantum BCH codes are constructed with even order.Factor code length with reasonable expression and then provide the necessary condition on cyclotomic cosets when BCH codes are dual-containing.Based on fundamental properties of cosets in classical encoding theory,suitable cyclotomic cosets are given.Compared with schemes available,this paper takes full advantage of Steane construction to generate new quantum codes.When focusing on Hermitian construction,q should be odd prime power or primitive quantum BCH codes can be obtained in known results.The dissertation designs primitive or non-primitive quantum BCH codes over any finite fields in our schemes,respectively.(3)The paper proposes constructions on quantum BCH codes in common sense.Construc-tions on q=2 or r=1 are generalized.Some lemmas on cosets are studied.In addition,various schemes are presented for different parameters.Not only are classical BCH codes able to construct new quantum BCH codes,but also it facilitates to compute the dimension of quantum codes.In particular,the defining set contains the most consecutive integers.There-fore,corresponding stabilizer codes have better lower bound of minimum distance.The quantum BCH codes in our thesis have better parameters than known results theoretically and practically.(4)Quantum codes are discussed from finite fields to finite rings,especially finite chain rings with special structure.This paper studies generating polynomial in terms of duality by employing polynomial decomposition and defining sets over finite chain rings.A necessary and sufficient condition for duality is presented.The dissertation proposes two schemes over finite chain rings and presents corresponding algorithms.Constructions on finite fields are extended to finite chain rings.The gap is filled in quantum error-correcting codes over finite chain rings.The paper discusses the relationships on quantum BCH codes between finite fields and finite rings roughly.The problem of finding quantum-error-correcting codes is transformed into the problem of finding quantum codes over Galois rings.