Y.S.Han Cyclic codes Consider the polynomial xg(x),22g(x),...,xm--1g(x). Clearly,they are cyclic shifts of g(x)and hence code polynomials in C.Since C is linear,a linear combination of g(x),xg(x),...,xn-r-1g(x), v(x）=uog(r)+u1xg(）+…+n-r-1an-r-1g(x) =(o+w1x+·+n-r-1xn-T-1)g(x), is also a code polynomial where ui∈{0，l}. Let g(z)=1+gx+...+gr-1x"-1+x"be the nonzero code polynomial of minimum degree in an (n,k)cyclic code C.A binary polynomial of degree n-1 or less is a code polynomial if and only if it is a multiple of g(x). Proof:Let v(x)be a binary polynomial of degree n-1 or less. Suppose that v(x)is a multiple of g(x).Then v(x)=(ao+az+...+an-r-1x"-r-1)g(x) School of Electrical Engineering Intelligentization,Dongguan University of TechnologyY. S. Han Cyclic codes 5 • Consider the polynomial xg(x), x2 g(x), . . . , xn−r−1 g(x). Clearly, they are cyclic shifts of g(x) and hence code polynomials in C. Since C is linear, a linear combination of g(x), xg(x), . . . , xn−r−1 g(x), v(x) = u0 g(x) + u1 xg(x) + · · · + un−r−1 x n−r−1 g(x) = (u0 + u1 x + · · · + un−r−1 x n−r−1 )g(x), is also a code polynomial where ui ∈ {0, 1}. • Let g(x) = 1 + g1 x + · · · + gr−1 x r−1 + x r be the nonzero code polynomial of minimum degree in an (n, k) cyclic code C. A binary polynomial of degree n − 1 or less is a code polynomial if and only if it is a multiple of g(x). Proof: Let v(x) be a binary polynomial of degree n − 1 or less. Suppose that v(x) is a multiple of g(x). Then v(x) = (a0 + a1 x + · · · + an−r−1 x n−r−1 )g(x) School of Electrical Engineering & Intelligentization, Dongguan University of Technology