Research On Linear Codes Under The Symbol-pair Metric

Based on linear algebra,abstract algebra,finite field,combinatorial theory and algebraic coding theory,we studied the algebraic properties of linear codes over finite fields under the b-symbol metric,the b-symbol weight distribution of several classes of cyclic codes,and parameters of some cyclic codes over finite fields in this Dissertation.Our specific research contents are as follows:1.The relations and differences among Hamming metric,generalized Hamming metric and b-symbol metric are studied.For the first time,we use the run in the sequence to calculate the b-symbol weight of a vector.For any vector c,we show the relationship between the Hamming weight and the b-symbol weight of c.Based on this,we carefully compared the three minimum distances of a linear code C under three metrics(Hamming metric,generalized Hamming metric,b-symbol metric)and analyzed the similarities and differences.2.In this chapter,we study several classes of irreducible cyclic codes over finite fields and give the complete b-symbol weight distribution of such codes.The formula for calculating the b-symbol weight of arbitrary irreducible cyclic codes over finite fields is studied.Several classes of cyclic codes with few b-symbol weights are constructed.Using the characteristics of b-symbol read channel,we further give the optimal codes construction for the shortened codes.3.In this chapter,we study a class of reducible cyclic codes C(Kasami codes)over finite fields and give the complete symbol-pair weight distribution of this class of reducible cyclic codes.Considering the algebraic stucture of Kasami codes,we discuss them in four cases and obtain relevant results.We give two applications of the b-symbol weight distribution of Kasami codes.4.The properties and parameters of antiprimitive BCH codes over finite fields are studied.The dimension and minimum distance of these antiprimitive BCH codes can be determined in some cases.We gave some lower bounds on the minimum distance of this class of antiprimitive BCH codes and partially solve two open problems related to them.5.Several classes of cyclic codes generated by low differentially uniform functions are constructed and their parameters and applications are studied.We have supplemented and corrected the previous results.We also solved(or partially solved)several open problems related to them.