Quantum codes from cyclic codes over F₃ + vF₃

Abstract

Let R = F3 + vF3 be a finite commutative ring, where v2 = 1. It is a finite semi-local ring, not a chain ring. In this paper, we give a construction for quantum codes from cyclic codes over R. We derive self-orthogonal codes over F3 as Gray images of linear and cyclic codes over R. In particular, we use two codes associated with a cyclic code over R of arbitrary length to determine the parameters of the corresponding quantum code.

On Quantum Codes From Cyclic Codes Over a Class of Nonchain Rings

“…In this study we extend the findings obtained in [3] to a more general class of nonchain rings R p = F p + vF p + · · · + v p−1 F p , where v p = v and p is a prime. Firstly, we state the Gray map defined in [4] and its properties and reprove that this Gray map preserves the orthogonality from R n p to F pn p .…”

“…To give the exact parameters of a quantum code constructed via CSS construction, it is enough to find a classical linear code containing its dual. Hence, the conditions for linear codes containing their duals have been investigated in [1,3,7,12,16]. In [1], the BCH codes containing their Euclidean or Hermitian duals were studied and the quantum BCH codes were obtained.…”

“…By taking the Gray images of some special cyclic codes over F 2 + uF 2 , a class of binary quantum codes were obtained. In [3], the findings in [12] are generalized to the ring F 3 + vF 3 with v 2 = 1 and a construction of a class of ternary quantum codes is presented. Further, in [7], the conditions of cyclic codes over the ring…”

On the Codes over a Semilocal Finite Ring

“…Many good quantum codes have been constructed by using classical cyclic codes over finite fields or finite rings with self orthogonal (or dual containing) properties in [3,12,13,16,19,24,25,28,29,31].…”

“…In [4] they introduced the finite ring R = Z 3 [v]/ v 3 − v . They studied the structure of this ring.…”