Executive Summary : | Gao et al. studied a class of generalized quasi-cyclic (GQC) codes and introduced skew GQC codes in 2016. Under factorization of ideals, they proved the Chinese Remainder Theorem for the skew polynomial ring R= Fq[x,α], which leads to a canonical decomposition of skew GQC codes and some characteristics of ρ-generator skew GQC codes. But, their work was under the restriction that the order of α divides each mi, i = 1, 2, …, t. If the restriction is removed, then the polynomial xm-1 is not a central element, i.e., the set R/(xm -1) is not a ring. In this case, the cyclic code in R/(xm -1) will not be an ideal. It is just a left R-submodule, and we call it a module skew cyclic code. Precise, it is still open to investigate the skew cyclic codes over finite rings without the restriction that the order of α divides each mi, i = 1, 2, …, t. The specific problems to study in this project are as follows: 1. By dropping the above restriction on α, we study the structural properties of module skew cyclic codes and module skew GQC codes over finite fields. 2. The Main motive is to find some new and optimal codes over finite fields from the skew GQC codes. 3. Later, we attempt to find some new and good quantum codes from the skew GQC codes. |