Extended Cyclic Codes Sandwiched Between Reed–Muller 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 Codes

A relation between quasi-cyclic codes and 2-D cyclic codes

We consider a q-ary quasi-cyclic code C of length m` and index `, where both m and ` are relatively prime to q. If the constituents of C are cyclic codes, we show that C can also be viewed as a 2-D cyclic code of size m × ` over Fq. If we further assume that m and ` are also coprime to each other, then we easily observe that the code C must be equivalent to a cyclic code. The last fact was prov...

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...