Constacyclic Codes And Their Applications In Entanglement-assisted Quantum Codes

With the rapid development of quantum communication,quantum error correction has become a new research hotspot.In the research of quantum error-correcting codes,it is found that if the encoder and the decoder share the entanglement state in advance.Then any classical codes can be used to construct the entanglement assisted quantum codes.In other words,this allows arbitrary classical codes to be quantized.Constacyclic codes over finite fields are widely used in algebraic coding because they are easy to encode and easy to decode.Therefore,in this paper,we construct eight classes of entanglement-assisted quantum codes with flexible parameters and one class of entanglement-assisted quantum codes with a certain number of entangled states,using cyclic codes and negacyclic codes in finite fields as source codes.Firstly,entanglement-assisted quantum codes are constructed by the cyclic codes with length n=(q2+1)/5 over finite field Fq2.Then,we give another representation of the q2-cyclotomic cosets of module n under the four cases of q=10θ+3(θ≥ 2),q=10θ+7(θ≥ 2),q=2e(e ≡ mod 4)and q=2e(e≡3 mod 4)respectively.Next,the number of entangled states in different cases is determined successfully by decomposition and selection of appropriate defining sets.Furthermore,four classes of entanglement-assisted quantum codes are constructed.According to the Singleton bound for the entanglement-assisted quantum codes,these four codewords we constructed are entanglement-assisted quantum MDS codes.Secondly,entanglement-assisted quantum codes are constructed by the negacyclic codes with length n=(q2+1)/13 over finite field Fq2.Then,we give another representation of the q2-cyclotomic cosets of module 2n containing the odd number from n/2 to 3n/2 under the four cases of q=26θ+5(θis positive even),q=26θ+5(θis positive odd),q=26θ+21(θ is positive even)and q=26θ+21(θis positive odd)respectively.Next,the number of entangled states in different cases is determined successfully by decomposition and selection of appropriate defining sets.Furthermore,four families of entanglement-assisted quantum codes are constructed.According to Singleton bound for the entanglement-assisted quantum codes,these four codewords are entanglement-assisted quantum MDS codes.Finally,entanglement-assisted quantum MDS codes are constructed by taking the negacyclic codes of length n=(q2-1)/12 over a finite field Fq2 as the code source.The q2-cyclotomic cosets Ci of the module 2n containing the odd number i(1≤i≤2n)is calculated at first.Then the dual containing condition is given.Furthermore,a class of entanglement-assisted quantum codes with a certain number of entangled states is constructed by selecting an appropriate definition set.According to Singleton bound for the entanglement-assisted quantum codes,this kind of codeword is the entanglement-assisted quantum MDS codes.