Research On Generalized Reed-Solomon Codes And Constacyclic Codes Over Finite Fields And Their Application

The theory of classical error-correcting codes is an important guarantee of traditional digital communication.Linear codes over finite fields are an important part of classical error-correcting codes.Generalized Reed-Solomon codes and constacyclic codes are two kinds of linear codes with good algebraic structure,which play an important role in the theory of error-correcting codes.Similarly,quantum error-correcting codes(quantum codes)are an important guarantee for quantum computing and quantum communication.There are essential differences between quantum codes and classical codes,but some quantum codes with good parameters can be constructed by using classical linear codes.Generalized Reed-Solomon codes and constacyclic codes over finite fields are studied in this paper.By using Hermitian self-orthogonal generalized Reed-Solomon codes,we construct two new classes of q-ary quantum maximum-distance-separable codes(quantum MDS codes),which have minimum distance greater than q/2.Many classes of entanglement-assisted quantum MDS codes are constructed by generalized Reed-Solomon codes.By using constacyclic codes,we construct some new classes of q-ary entanglement-assisted quantum MDS codes with length(q2+1)/13.Finally,optimal cyclic locally repairable codes(LRCs)are constructed by using cyclic codes.The details are as follows:(1)We construct two classes of Hermitian self-orthogonal generalized Reed-Solomon codes,and use these two classes of generalized Reed-Solomon codes to construct two new classes of q-ary quantum MDS codes with minimal distance greater than q/2.These two classes of quantum MDS codes have new parameters.(2)We calculate the dimensions of many classes of generalized Reed-Solomon codes and extended generalized Reed-Solomon codes.We also construct many classes of entanglement-assisted quantum MDS codes according to these codes.By comparison,it can be found that most of the entanglement-assisted quantum MDS codes have new parameters.(3)We study the decomposition of definition sets of some constacyclic codes and cyclic codes of length(q2+1)/1/3.Four classes of q-ary entanglement-assisted quantum MDS codes with length(q2+1)/13 are constructed by using these constacyclic codes and cyclic codes.Through a lot of comparisons,it is shown that the parameters of the entanglement-assisted quantum MDS codes constructed in this paper are different from those constructed in other literatures.(4)We constructed six classes of optimal cyclic(r,δ)-LRCs with unbounded length and minimum distance δ+2≤d≤2δ from their zeros.Two classes of optimal cyclic(r,δ)-LRCs with large minimum distance are also presented.These LRCs with unbounded length can be regarded as an extension of two kinds of existing constructs.