Y.S.Han Cyclic codes 2 Proof:Multiplying v(x)by x,we obtain x2v()=0x2+v1x+1+…+n--1xn-1++un-1an+i-1. Then we manipulate the equation into the following form: x2v(x)=n-i+n-i+1x+…+n-1xi-1+0x2+. +n-i-1xn-1+vn-i(xn+1)+wn-i+1x(x”+1) +…+vn-1x2-1(xn+1) =q(x)(xn+1)+v@(c), Where q(x)=Un-i+Un-i+1x+...+Un-1xi-1, The nonzero code polynomial of minimum degree in a cyclic code C is unique. Let g(x)=go+gx+...+gr-1x"-1+z"be the nonzero code polynomial of minimum degree in an (n,k)cyclic code C. Then the constant term go must be equal to 1. School of Electrical Engineering Intelligentization,Dongguan University of Technology Y. S. Han Cyclic codes 2 Proof: Multiplying v(x) by x i , we obtain x i v(x) = v0 x i + v1 x i+1 + · · · + vn−i−1 x n−1 + · · · + vn−1 x n+i−1 . Then we manipulate the equation into the following form: x i v(x) = vn−i + vn−i+1x + · · · + vn−1 x i−1 + v0 x i + · · · +vn−i−1 x n−1 + vn−i (x n + 1) + vn−i+1x(x n + 1) + · · · + vn−1 x i−1 (x n + 1) = q(x)(x n + 1) + v (i) (x), where q(x) = vn−i + vn−i+1x + · · · + vn−1 x i−1 . • The nonzero code polynomial of minimum degree in a cyclic code C is unique. • Let g(x) = g0 + g1 x + · · · + gr−1 x r−1 + x r be the nonzero code polynomial of minimum degree in an (n, k) cyclic code C. Then the constant term g0 must be equal to 1. School of Electrical Engineering & Intelligentization, Dongguan University of Technology