Six Classes Of Q-ary Cyclic Codes And Their Parameters

Due to their efficient encoding and decoding algorithms,cyclic codes have been used in in communications,information security,and data storage systems.Thanks to their good algebraic and combinatorial structures,cyclic codes have close relation with many mathemat-ical subjects such as matrices,exponential sums,difference sets,t-designs,polynomials,and Graph.During the last six decades,finding cyclic codes with excellent parameters has been an interesting and hot topic in coding theory and receive a lot of attentions.Polynomials and matrices over finite fields are two main powerful tools for constructing cyclic codes meeting or approaching the theorectial bounds.Inspired the idea of Ding and Helleseth for construcing ternary cyclic codes,we con-struct and study six classes of q-ary cyclic codes using well designed generator polynomials over finite fields.The selected polynomials are of the form M?u(x)M?v(x),where q is an odd prime,? is a primitive element of Fq~m,gcd(m,q-1)=1,and M?i(x)denotes the minimal polynomial of ?i over IFq.The correspoding exponents u,v for the six class of cyclic codes are respectively given by1.u=q-1,v=(q~k+1)(q-1),where gcd(m,k)=1;2.u=q-1,v=(q~k+1)(q-1)/2,where gcd(m,k)=1;3.u=q-1,v=(q(2t+1)k+1)(q-1)/q~k+1,where gcd(t(t+1)k,m)=1;4.u=q-1,v=(q(t+1)k-1)(q-1)/q~k-1,where gcd(t(t+1)k,m)=1;5.u=q-1,v=r(q-1),where gcd(m,r!)=1;6.u=q-1,v=q-1/2.Based on the theory of polynomials over finite fieds,we prove the first five classes of cyclic codes have parameters[q~m-1/q-1,q~m-1/q-1-2m,b],4<d<5,and the last class has parameters[2(q~m-1)/q-1,2(q~m-1)/q-1)-2m,d],4?d?5.We also prove these six classes of cyclic codes are optimal or almost optimal with respect to a bound on linear code.The first class of the proposed cyclic codes include an earlier one as a special case.