Research On Cyclic Self-orthogonal Codes Over Finite Rings

Self-orthogonal codes over finite rings or fields are a class of important linear codes, which play a significant role in coding theory. Especially, self-dual codes have been an important subject in the research of error-correcting codes. With a deeply study in quantum error-correction, it has been found that classical self-orthogonal codes can be used construct quantum error-correcting codes. This has caused the great interest in constructing classical self-orthogonal codes.In this paper, we study cyclic self-orthogonal codes over the finite ring Fq+uFq and give a method of constructing self-orthogonal codes over the finite field Fq. The paper includes two parts:On one hand, a method is proposed to construct self-orthogonal codes over the finite field Fq by using cyclic self-orthogonal codes over the finite ring Fq+uFq. A Gray map from Fq+uFq to Fqp is introduced, and a sufficient and necessary condition for the existence of cyclic self-orthogonal codes over Fq+uFq is given. Further, some self-orthogonal codes over Fq with good parameters are constructed.On the other hand, the relationship between cyclic codes and (1+u)-constacyclic codes over the ring F2-+F2- is studied. The Gray images of (1+u)-constacyclic codes over F2-+uF2- are determined by using a map from Fr+uF2. to F2-2. Further, it is proved that cyclic self-dual codes of any odd length over F2-+uF2. are Type I codes.