DNA Codes Over Two Classes Of Finite Sets

In recent years,some scholars have used DNA calculations to solve some NPC problems and found that they had unprecedented super-large-scale parallel computing capabilities,which will trigger a new information revolution.How to design a DNA code with good algebraic structure and easy for coding and decoding is a basic problem in DNA computing.At present,many scholars have constructed DNA codes by using classical error correcting codes such as linear codes,cyclic codes,Goppa codes,and BCH codes.In this dissertation,we mainly study DNA codes over two classes of finite sets.The details are as follows:On the one hand,DNA codes over the finite fieldF_4^k are studied.We define m-quasi-self-reciprocal polynomials over the finite field F_q firstly,and obtain some optimal reversible codes over the finite fields F_q by using m-quasi-self-reciprocal polynomials.Further,we define DNA-m-quasi-self-reciprocal polynomials over the finite field F_q,and a one-to-one correspondence between DNA sequence of length k and every element over the finite fieldF_4^k is given,then reversible DNA codes and reversible complement DNA codes over the finite fieldF_4^k are obtained by using DNA-m-quasi-reciprocal polynomials.On the other hand,we study DNA codes over the finite ringF₄[v]/（V²+v）,and obtained a necessary and sufficient condition of DNA codes over the finite ringF₄[v]/（V²+v）to be reversible（complementary）DNA codes.