In this paper we introduce self-dual cyclic and quantum codes over Zα2 × (Z2 + uZ2) . We determine the conditions for any Z2Z2[u]-cyclic code to be self-dual, that is, C = C. Since the binary image of a selforthogonal Z2Z2[u]-linear code is also a self-orthogonal binary linear code, we introduce quantum codes over Zα2 × (Z2 + uZ2) β . Finally, we present some examples of self-dual cyclic and qu...

Let F2m be a finite field of characteristic 2 and R = F2m [u]/〈u 〉 = F2m + uF2m + . . . + u k−1 F2m (u k = 0) where k ∈ Z satisfies k ≥ 2. For any odd positive integer n, it is known that cyclic codes over R of length 2n are identified with ideals of the ring R[x]/〈x − 1〉. In this paper, an explicit representation for each cyclic code over R of length 2n is provided and a formula to count the n...

A binary linear code C is a Z2-double cyclic code if the set of coordinates can be partitioned into two subsets such that any cyclic shift of the coordinates of both subsets leaves invariant the code. These codes can be identified as submodules of the Z2[x]-module Z2[x]/(x r −1)×Z2[x]/(x − 1). We determine the structure of Z2-double cyclic codes giving the generator polynomials of these codes. ...

A Gray code is said to be cyclic if the last element and the first element in the sequence also differ by some pre-specified constant amount. The Gray code above demonstrates the cyclic property. It is well known that a cyclic Gray code of binary string corresponds to a Hamilton cycle in the n-dimensional hypercube. Figure 1 shows the 3-dimensional hypercube Q3 with the Hamilton cycle correspon...

We prove, without assuming the Generalized Riemann Hypothesis, but with at most one exception, that an irreducible cyclic code c(p, m, v) with v prime and p of index 2 modulo v is a two-weight code if and only if it is a semiprimitive code or it is one of the six sporadic known codes. The result is proved without any exception for index-two irreducible cyclic c(p, m, v) codes with v prime not c...

A Z2Z4-additive code C ⊆ Z α 2 ×Z 4 is called cyclic if the set of coordinates can be partitioned into two subsets, the set of Z2 and the set of Z4 coordinates, such that any cyclic shift of the coordinates of both subsets leaves the code invariant. We study the binary images of Z2Z4-additive cyclic codes. We determine all Z2Z4-additive cyclic codes with odd β whose Gray images are linear binar...

A (cyclic) n-bit Gray code is a (cyclic) ordering of all 2 binary strings of length n such that consecutive strings differ in a single bit. Equivalently, an nbit Gray code can be viewed as a Hamiltonian path of the n-dimensional hypercube Qn, and a cyclic Gray code as a Hamiltonian cycle of Qn. In this paper we study Hamiltonian paths and cycles of Qn avoiding a given set of faulty edges that f...

A (cyclic) n-bit Gray code is a (cyclic) ordering of all 2 binary strings of length n such that consecutive strings differ in a single bit. Equivalently, an n-bit Gray code can be viewed as a Hamiltonian path of the n-dimensional hypercube Qn, and a cyclic Gray code as a Hamiltonian cycle of Qn. In this paper we study (cyclic) Gray codes avoiding a given set of faulty edges that form a matching...

In this paper, we have studied cyclic codes over the ring R = Z4 +uZ4, u = 0. We have considered cyclic codes of odd lengths. A sufficient condition for a cyclic code over R to be a Z4-free module is presented. We have provided the general form of the generators of a cyclic code over R and determined a formula for the ranks of such codes. In this paper we have mainly focused on principally gene...

In analogy to cyclic codes, we study linear codes over finite fields obtained from left ideals in a quotient ring of a (non commutative) skew polynomial ring. The paper shows how existence and properties of such codes are linked to arithmetic properties of skew polynomials. This class of codes is a generalization of the θ-cyclic codes discussed in [1]. However θ-cyclic codes are performant repr...