Research On Some Linear Codes Over Finite Commutative Rings

Coding theory over finite commutative rings has become one of the most important branches of coding fields. Many researchers devote to study the linear codes over these rings. In their studies, structure properties of linear codes, constructing of good codes and the decoding problem have occupied the main research areas. And there are some inspring resultes in their reseach.In this paper, we mainly study some important linear codes over finite commutative rings, including cyclic codes over finite chain rings and1-generator quasi-cyclic codes over Galois rings, cyclic codes over commutative but no chian ring Fq+uFq+vFq+uvFq and (l+u)-constacyclic codes over F2m+uF2m+vF2m+uvF2m, constructing quai-cyclic codes and negaquai-cyclic codes over finite fields.Firstly, we research the linear codes over finite chain rings. In this section, we mainly research the cyclic codes over finite chian rings and1-generator quasi-cyclic codes over Galois rings. For cyclic codes, we survey their stucture properties. And for quasi-cyclic codes, we give the structure of the annihilators of1-generator quasi-cyclic codes. Then, we discuss the enumeration and the generators of1-generator quasi-cyclic codes.Secondly, we mainly study the cyclic codes over the commutative but no chain ring Fq+uFq+vFq+uvFq. And then, we study the (1+μ)-constacyclic codes over F2m+uF2m+vF2m+uvF2m. We have that under the Gray map Φ, the image of (l+u)-constacyclic code with length n is a cyclic code over F2m+uF2m with length2n.Finally, we study the quasi-cyclic codes and2-generator negaquasi-cyclic codes over finite fields. In this section, we mainly devote to construct good codes over finite fields. We give two construction methods. And by these, we get some good linear codes over finite fields.