Y.S.Han Cyclic codes Suppose that the message to be encoded is u=(uo,u1,...,uk-1).Then xn-ku(c)=uoxn-k+u1xn-k+1+…十uk-1xn-1. Dividing z-ku(x)by g(x),we have x"-ku(x)=a(x)g(z)+b(x). Since the degree of g(x)is n-k,the degree of b(x)must be n-k-1 or less.Then b(x)+x"-ku(x)=a(x)g(x) is a multiple of g(x)and therefore it is a code polynomial. b(c）+xn-u(c）=b0+b1x+…+bn-k-1xn-k-1 +uoxn-k+u1x”-k+1+…+-1xn-1 School of Electrical Engineering Intelligentization,Dongguan University of TechnologyY. S. Han Cyclic codes 9 • Suppose that the message to be encoded is u = (u0, u1, . . . , uk−1). Then x n−k u(x) = u0 x n−k + u1 x n−k+1 + · · · + uk−1 x n−1 . Dividing x n−k u(x) by g(x), we have x n−k u(x) = a(x)g(x) + b(x). Since the degree of g(x) is n − k, the degree of b(x) must be n − k − 1 or less. Then b(x) + x n−k u(x) = a(x)g(x) is a multiple of g(x) and therefore it is a code polynomial. b(x) + x n−k u(x) = b0 + b1 x + · · · + bn−k−1 x n−k−1 +u0 x n−k + u1 x n−k+1 + · · · + uk−1 x n−1 School of Electrical Engineering & Intelligentization, Dongguan University of Technology