Three Questions About Quasi-cyclic Codes And Constacyclic Codes

Coding theory studies mainly the mathematical structure of codes and con-structing good codes.Cyclic codes are an extremely important class of codes.From the point of view of constructing good codes,a long standing open question is:are the cyclic codes asymptotically good?A classical result is:the quasi-cyclic codes of index 2 are asymptotically good.The duality plays an important role in coding theory,which has important applications in the researches of weight structure and algebraic structure.This dissertation is focusing on three topics around cyclic codes and its generalization.1.Quasi-cyclic codes of fractional index.We first introduce quasi-cyclic codes of fractional index,study their algebraic structure,and prove that:the quasi-cyclic codes of fractional index between 1 and 2 are asymptotically good.We extend this research idea to Z2Z4-additive cyclic codes,obtain the generator matrices of a class of Z2Z4-additive cyclic codes,and prove that Z2Z4-additive cyclic codes are also asymptotically good.To study the algebraic structure of the two kinds of codes,double circulant matrices are introduced,the rank r of a double circulant matrix is computed,and it is shown that any consecutive r rows of the double circulant matrix are linearly independent.2.Galois LCD codes over finite fields.Generalizing LCD codes and Hermitian LCD codes,we introduce k-Galois LCD codes.We obtain necessary and sufficient conditions for linear codes and constacyclic codes to be k-Galois LCD codes,and show a method of constructing such k-Galois LCD codes.Some classes of k-Galois LCD MDS codes are obtained.3.Self-dual constacyclic codes over finite rings.When some quotient rings are not chain rings,it is difficult to determine self-dual repeated-root constacyclic codes over finite commutative chain ring Fpm +uFpm,there are few results in that case.By studying dual codes and the structure of solutions of matrix equations,we determine completely all self-dual constacyclic codes of length ps and 2ps over Fpm + uFpm.