On Cyclic Codes Over Rings And Their Applications

With the development of coding theory,coding theory over finite rings has been becoming more and more important from theory to application.Many domestic and overseas coding theorists devote themselves to this topic.In this paper,we study the relationship between the cyclic code sequences and the DNA base sequences by studying the theories of cyclic codes over rings,and obtained cyclic DNA codes with excellent performance.Based on the previous excellent work on cyclic DNA codes over finite rings,the thesis makes a further research on cyclic DNA codes over finite chain ring F₄[u]/?u²+1?and finite non-chain ring Z₄+?Z₄.The details are given as follows:In Chapter 2,we introduce basic theories of cyclic codes and give some necessary algebraic knowledge and results on coding theory.In Chapter 3,we mainly study the theories of cyclic codes over finite chain ring F₄[u]/?u²+1?,and establish the 1-1 correspondence between the elements over the ring and the DNA base pairs,and construct the cyclic DNA codes over the ring.Because DNA molecules are disordered and complementary,we have studied the structure and properties of cyclic DNA codes over the ring.At the same time,the generator of cyclic DNA codes over the ring is given.Finally,using the above results,we construct a cyclic DNA code with length 6 satisfying reflexive and complement.In Chapter 4,we mainly study the theories of cyclic codes over finite non-chain ring R=Z₄+?Z₄.By defining a Gray map between R and Z₄he relationship between the cyclic codes over Z₄ and the cyclic codes with arbitrary code lengths over R is established.The generator of the cyclic codes over R is obtained.Using the developed Gray map,the relationship between the elements over R and the DNA base pairs is established,and the cyclic DNA codes of R is constructed.Moreover,the reversible constraints,reversible complementary constraints and GC weight constraints of the cyclic DNA codes over R are also studied,and the generator of cyclic DNA codes over R is also given.Finally,using the above results,we construct the cyclic DNA codes with the code lengths of 8 and 10,respectively.