ΘS-cyclic codes overAk

Cyclic Codes

Polynomial Codes and Cyclic Codes

As an example, take F2, n = 5 and g(x) = x2 + x + 1. The code consists of the 8 codewords 0 · g(x), . . . , (x2 + x + 1) · g(x). Equivalently, we can identify every polynomial with its vector of coefficients to get a codeword in F2 . Verify that a polynomial code is linear and has dimension k = n −m. Also, check that if g(x) = ∑n−k i=0 gix i is the generator polynomial, then an n× k generating ...

Cyclic Sieving for Cyclic Codes

These are notes on a preliminary follow-up to a question of Jim Propp, about cyclic sieving of cyclic codes. We show that two of the Mahonian polynomials are cyclic sieving polynomials for certain Dual Hamming Codes: X and X inv for q = 2, 3 and q = 2, respectively.

On quasi-cyclic codes as a generalization of cyclic codes

In this article we see quasi-cyclic codes as block cyclic codes. We generalize some properties of cyclic codes to quasi-cyclic ones such as generator polynomials and ideals. Indeed we show a one-to-one correspondence between l-quasi-cyclic codes of length lm and ideals of Ml(Fq)[X]/(X m −1). This permits to construct new classes of codes, namely quasi-BCH and quasi-evaluation codes. We study th...

Cyclic additive codes and cyclic quantum stabilizer codes

The theory of cyclic linear codes in its ring-theoretic formulation is a core topic of classical coding theory. A simplified approach is in my textbook [1]. The language of ring theory is not needed. We will present a self-contained description of the more general theory of cyclic additive codes using the same method. This includes cyclic quantum stabilizer codes as a special case. The basic in...