Quadratic Residue Codes And Support Weight Distribution Over Finite Non-chain Rings

Cyclic codes have a very important position in the theory of error correc-tion codes, and quadratic residue codes is a kind of important cyclic codes, a lot of good performance of error correction codes comes from quadratic residue codes. Gleason first gives the definition of quadratic residue codes, from then on, people began persistent research in this field. In this thesis, we mainly investigate quadratic residue codes and support weight distribution over the finite non-chain rings. This thesis is divided into three parts.Chapter 1, we will briefly introduce the quadratic residue codes and weight distribution of the rings of related background knowledge, as well as some knowledge of the involved with the main work in this paper.In the second chapter, quadratic residue codes over Fp+uFp+vFp+uvFp are investigated in this article, where u2=u,u2= v,uv= vu and p is odd prime. At first, the basic knowledge of cyclic codes of length n is introduced over Fp+uFp+vFp+uvFp, then the structures of quadratic residue codes over Fp+uFp+vFp+uvFp are obtained by applying their generating idempotents, and the relations and dual properties between these codes and their extended codes are discussed. Finally, the specific forms of generating idempotents of quadratic residue codes over the ring are given.In the third part, we mainly introduce the support weight distribution over Fp+vFp+v2Fp and Fp+uFp+vFp+uvFp, the relation between the support weight distribution of a linear code of type p4k and p3k and the Hamming weight distribution were established, and the relation between the support weight distribution of the linear code and its dual was derived by MacWilliams-identity over the rings.