Research On Some Problems Of Two Classes Of Generalized Cyclic Codes

Along with further improvement of coding theory on finite fields, and after that error correcting-codes over finite rings have been given more attention of scholars.In this paper, we mainly study the structure and their associated properties of constacyclic codes and skew cyclic codes over two classes of finite rings. The details are given as follows:1. Firstly, we study the classification of cyclic codes of length 3ps over the ring R= Fpm+uFpm, and give the generator polynomials. Secondly, discuss the enumeration of the cyclic codes, and give its annihilator under different conditions. Finally, we study the classification of constacyclic codes of length 3ps over the ring R= Fpm+uFpm according to the Chinese remainder theorem, we prove:(1) when pm=-1(mod3), code C is a (α+uβ)-constacyclic code of length3ps over R if and only if code C is a (1+uθ)-constacyclic code of length 3ps over R. Code C is a γ-constacyclic code of length 3ps over R if and only if code C is a cyclic code of length 3ps over R.(2) when pm=1(mod3),code C is a (λ1,+βu)-constacyclic code of length 3ps over R if and only if code C is a(ωl-λl-ωl βu)-constacyclic code of length 3ps over R. Code C is a λl-constacyclic code of length 3ps over R if and only if code C is a ωl-constacyclic code of length 3ps over R.2. The skew polynomial rings and skew cyclic codes over ring R=Z4+uZ4 are defined. Based on the structural properties of skew polynomial ring, we proved that the skew cyclic code of length n over R is a left R[x;σ]-submodule of R[x;σ]/(xn-1). Finally, the generator polynomials and definitions of the duals of skew cyclic codes with respect to Euclidean and Hermitian inner products are given.