Y.S.Han Cyclic codes Description of Cyclic Codes If the components of an n-tuple v=(vo,v1,...,Un-1)are cyclically shifted i places to the right,the resultant n-tuple would be v）=(n-i,vn-i+1,,vn-l,0,U1,,n-i-1). Cyclically shifting v i places to the right is equivalent to cyclically shifting v n-i places to the left. An (n,k)linear code C is called a cyclic code if every cyclic shift of a code vector in C is also a code vector in C. Code polynomial v(x)of the code vector v is defined as u(x)=0+U1x+…+n-1xn-1. ·v((c)=xv(x)）mod zm+1. School of Electrical Engineering Intelligentization,Dongguan University of TechnologyY. S. Han Cyclic codes 1 Description of Cyclic Codes • If the components of an n-tuple v = (v0, v1, . . . , vn−1) are cyclically shifted i places to the right, the resultant n-tuple would be v (i) = (vn−i , vn−i+1, . . . , vn−1, v0, v1, . . . , vn−i−1). • Cyclically shifting v i places to the right is equivalent to cyclically shifting v n − i places to the left. • An (n, k) linear code C is called a cyclic code if every cyclic shift of a code vector in C is also a code vector in C. • Code polynomial v(x) of the code vector v is defined as v(x) = v0 + v1 x + · · · + vn−1 x n−1 . • v (i) (x) = x i v(x) mod x n + 1. School of Electrical Engineering & Intelligentization, Dongguan University of Technology