Research And Construction Of Cyclic Codes Based On Q-polynomial Approach

Cyclic codes is an important class of linear codes,which also have important application in the fields of electronic and communication.It is of great significance to the research and construction of cyclic codes.In recent years,experts and schloars have carried out a lot of in-depth research on cyclic codes,and have obtained a large number of outstanding research results.In 2013,Cunsheng Ding and other scholars first proposed the q-polynomial approach for researching and constructing cyclic codes.This new approach is very useful for researching and constructing cyclic codes.This paper will further research and construct good cyclic codes based on the q-polynomial approach,and define another q-polynomial approach to cyclic codes.The results of the research are as follows:1.Let p be a prime,q be a power of p,t be a positive integer,N be a positive divisor of qt-1 and(qf-1)+ N(qj-1),V1≤j≤t-1.By taking different N,we construct several classes of cyclic codes with a length of n and a dimension of k.And the minimum Hamming distance of this kind of cyclic codes are classified,and several classes of almost optimal cyclic codes and optimal cyclic codes are cbtained.2.Let A is not a normal element of the vector space GF(qn)over GF(q),α is a normal element of the vector space GF(qn)over GF(q),for all b-(b0,b1,…,bn-1)∈ GF(q)n,ψλ:GF(q)n→GF(qn)b→n-1 ∑ i=0 biλqt,Im(ψλ)= {ψλ(b):b ∈ GF(q)n},Define C(n,α,ψλ)= {(c0,c1,…,cn-1)∈ GF(q)n:c =n-1 ∑ i=0 ciαqi ∈ Im(ψλ)},it’s a(A,a)q-polynomial code,and it is proved that C(n,α,ψλ)is a cyclic code of length n over GF(q),and every cyclic code C over GF(q)of length n can be expressed a code C(n,α,ψλ).