Research On Constacyclic Codes For Data Storage And Quantum Communication

One of the main themes in coding theory is constructing codes with good parameters to be applied with good performance.Different application environments have different require-ments for codes.As a coding scheme in high-density data storage systems,b-symbol codes are designed to overcome reading limitations in some typical scenarios.Locally repairable codes（LRCs）are designed for distributed storage systems to reduce repair costs and storage over-head.Asymmetric quantum error-correcting codes（AQECCs）protect quantum information from noise and decoherence,providing a necessary guarantee for quantum error corrections.Constacyclic codes are a significant class of codes with rich algebraic structures and have been applied by many scholars to construct codes with optimal parameters.In this thesis,we con-struct some b-symbol codes,locally repairable codes,and asymmetric quantum error-correcting codes with optimal parameters based on constacyclic codes.The specific research contents are as follows:（1）We characterize the b-symbol distances of a class of constacyclic codes of length np^sover a finite field F_q,where 1≤b≤np,p is the characteristic of the finite field,and n is coprime to p.The range of b we consider is larger than that of previous works.Maximum distance separable（MDS）b-symbol codes are b-symbol codes that achieve the Singleton bound for b-symbol distances with equality.Using the characterization of b-symbol distances,we obtain all MDS b-symbol codes among those constacyclic codes.Comparing code rates under the same b-symbol distances of MDS b-symbol codes,we conclude that our codes have larger code rates than the existing constructions in the literature.Furthermore,we generalize the results on the finite field F_qto a finite chain ring F_q+u F_qand obtain six classes of MDS b-symbol codes over F_q+u F_q.（2）Optimal LRCs that achieve the Singleton-type bound with equality are studied.For the single failed node case,we determine the locality of cyclic codes of length 2p^sover F_qand give a necessary and sufficient condition for such codes to be optimal.Furthermore,we characterize all the optimal cyclic LRCs of length 2p^sover F_q.These optimal LRCs possess new parameters.Moreover,some are simultaneously perfect LRCs achieving the Hamming-type bound.We give a new method for the multiple failed nodes case to analyze the（r,δ）-locality of cyclic codes.Via the characterization of locality,we obtain several classes of optimal LRCs of length p^s.When the minimum distance d is greater than or equal to 2δ+1,we present a class of optimal LRCs whose code lengths are unbounded.In contrast,the existing works only show the existence of optimal LRCs of code lengths O(q^1+δ2).Compared with the existing constructions,our codes do not have the restriction of d≤q.In other words,our optimal LRCs can have large distances over a relatively small field,which is desired by the practical requirement of low computation cost and high repair capability.（3）We construct new AQECCs by repeated-root cyclic codes with higher code rates than those in the literature.Some of these codes exceed the asymmetric quantum Gilbert-Varshamov（GV）bound.Furthermore,five classes of MDS AQECCs attaining the asymmetric quantum Singleton bound are constructed,and these codes have higher code rates than most of the ones obtained in the literature.