How to write cyclic codes?

1. How do you write a cyclic code?
2. How do you prove a code is cyclic?
3. Are cyclic codes linear codes?

How do you write a cyclic code?

It is straightforward to show that the observed subspace is cyclic if composed of polynomials divisible by a polynomial g(x) = g₀ + g₁x + … + g_n−kx^n−k that divides xⁿ − 1 at the same time. The polynomial g(x), of degree n − k, is called the generator polynomial of the code.

How do you prove a code is cyclic?

A polynomial code is cyclic if and only if its generator polynomial divides xn − 1. r(x) = −h(x)g(x) mod (xn − 1), so r(x) ∈ C. This means that r(x) = 0, since no other codeword in C can have degree smaller than deg(g).

Are cyclic codes linear codes?

A fundamental subclass of linear codes is given by cyclic codes, that enjoy a very interesting algebraic structure. In fact, cyclic codes can be viewed as ideals in a residue classes ring of univariate polynomials.