Quaternary 1-Generator Quasi-Cyclic Codes

Quaternary 1-generator quasi-cyclic codes are considered in the paper. Many scholars have worked on quasi-cyclic codes over finite fields, mainly focusing on three fields:study on the algebraic structure of quasi-cyclic codes; study on some quasi-cyclic codes with some good properties; study on decoding algorithm for quasi-cyclic codes. In these literatures,1-generator quasi-cyclic codes and their duals are the most frequently encountered quasi-cyclic codes, such as Bhargava, Seguin and Stein, Gulliver and Bhargava, Seguin have a lot of meaningful study in this field.In this paper, we present a one-to-one correspondence between quaternary 1-generator quasi-cyclic codes with the annihilator (f(x)g(x),2f(x)h(x))R and elements of the direct product (Sg(x)h(x)*/(Rg(x)h(x))*×(S f(x)h(x)*/(R f(x)h(x))*, under the conditions that n is odd and gcd(|2|n, m)=1(see Theorem 3.1.8). By the correspondence, we give the enu-meration of quaternary 1-generator quasi-cyclic codes, and describe an algorithm which will obtain one, and the only one, generator for each quaternary 1-generator quasi-cyclic code.