On Cyclic Codes Over Finite Rings And Their Applications

The cyclic code is an important type of linear code,which has the characteristics of error correction ability replacement and high error detection performance.And it is widely used in the error correction mechanism in actual communication.This paper studies the structural properties of cyclic codes over finite non-switching matrix ring M₂(F₂+uF₂),including:the algebraic structure of cyclic codes over finite non-switching matrix ring M₂(F₂+uF₂);the formation conditions of self-dual cyclic codes over finite non-switching matrix ring M₂(F₂+uF₂);The composition condition of the DNA code and the properties of the DNA code under the action of Gray mapping over finite non-switching matrix ring M2((F2+uF2).The specific research content is as follows.In Chapter 1,we mainly introduce the research background and research significance of cyclic codes on the finite non-switching matrix ring M₂(F₂+uF₂),the research status at home and abroad,and the summary of the main research results of this paper.In Chapter 2,we introduce the knowledge of algebra related to this article.We give the structure of the Gray map over the finite non-commutative matrix ring M₂(F₂+uF₂)and the algebraic structure of the cyclic code over the matrix ring M2((F2+uF2).And we also introduce the related knowledge of linear codes and cyclic codes over finite rings.In Chapter 3,we give the algebraic structure of the cyclic code over the matrix ring M₂(F₂+uF₂),and the algebraic structure of the self-dual cyclic code over the matrix ring M₂(F₂+uF₂).The structure of the cyclic code over the ring M₂(F₂+uF₂)constructs a self-dual code over finite field F₄,and gives the necessary and sufficient conditions for a self-dual code over ring M₂(F₂+uF₂).In Chapter 4,we study the structure of the cyclic DNA code over the finite non-commutative matrix ring M₂(F₂+uF₂).Under the action of Gray mapping,the reversible code and the reversible complement code of the upper cyclic DNA code are given over the ring M₂(F₂+uF₂).We use the base pairing in DNA to give the range of GC content that the cyclic DNA code makes the DNA code stable.In Chapter 5,the main results of the full paper are summarized,and on this basis,we further propose several problems to be solved urgently about cyclic codes over the finite non-switching matrix ring M₂(F₂+uF₂).