Proof

Let c = (c₀, ..., c_{n - 1}) C . Form the element c = c₀ + c₁X + ··· +c_{n - 1}X^{n - 1} Z/2Z[X].

Suppose c comes from the information vector a with corresponding polynomial a Z/2Z[X] of degree at most k - 1. Then c = ag + m for certain m (Xⁿ - 1). The degrees of c and of ag are at most n - 1. Therefore the degree of m is at most n - 1, too, and so m = 0. In particular, c = ag, and we obtain the following relation between c and a.

ch = agh = a·(Xⁿ - 1) = a - Xⁿa.

So a = ch mod Xⁿ.