c0 + c1x+ c2x 2 + · · ·+ cn−1x n−1 ∈ GF(q)[x]/(x − 1), any code C of length n over GF(q) corresponds to a subset of the quotient ring GF(q)[x]/(xn− 1). A linear code C is cyclic if and only if the corresponding subset in GF(q)[x]/(xn − 1) is an ideal of the ring GF(q)[x]/(xn − 1). It is well known that every ideal of GF(q)[x]/(xn−1) is principal. Let C = 〈g(x)〉 be a cyclic code, where g(x) is m...

We present a family of reducible cyclic codes constructed as the direct sum of two different semiprimitive two-weight irreducible cyclic codes. This family generalizes the class of reducible cyclic codes that was reported in the main result of B. Wang, et al. [11]. Moreover, despite of what was stated therein, we show that, at least for the codes studied here, it is still possible to compute th...

Abstract: Let Rk,m be the ring F2m [u1, u2, · · · , uk]/ ⟨ ui , uiuj − ujui ⟩ . In this paper, cyclic codes of arbitrary length n over the ring R2,m are completely characterized in terms of unique generators and a way for determination of these generators is investigated. A F2m -basis for these codes is also derived from this representation. Moreover, it is proven that there exists a one-to-one...

In this article, we count the quantity of minimal cyclic codes of length n and dimension k over a finite field Fq, in the case when the prime factors of n satisfy a special condition. This problem is equivalent to count the quantity of irreducible factors of x − 1 ∈ Fq[x] of degree k.

We introduce linear cyclic codes over the ring F 2 + uF 2 = f0; 1; u; u = u + 1g, where u = 0 and study them by analogy with the Z4 case. We give the structure of these codes on this new alphabet. Self-dual codes of odd length exist as in the case of Z4-codes. Unlike the Z4 case, here free codes are not interesting. Some nonfree codes give rise to optimal binary linear codes and extremal self-d...

Recently, a new algorithm to test equivalence of two cyclic codes has been introduced which is efficient and produced useful results. In this work, we generalize constacyclic codes. As an application the found many with good parameters properties. particular, 22 that improve minimum distances best known linear (BKLCs).

We propose two types, namely Type-I and Type-II, quantum stabilizer codes using quadratic residue sets of prime modulus given by the form p = 4n ± 1. The proposed Type-I stabilizer codes are of cyclic structure and code length N = p. They are constructed based on multi-weight circulant matrix generated from idempotent polynomial, which is obtained from a quadratic residue set. The proposed Type...

We study -length Abelian codes over Galois rings with characteristic , where and are relatively prime, having the additional structure of being closed under the following two permutations: i) permutation effected by multiplying the coordinates with a unit in the appropriate mixed-radix representation of the coordinate positions and ii) shifting the coordinates by positions. A code is -quasi-cyc...