QUANTUM CODES OBTAINED THROUGH (1+(p−2)ν)-CONSTACYCLIC CODES OVER Zp +νZp

Abstract. This paper is concerned with, structural properties and construction of quantum codes over Zp by using (1+(p−2)ν)-Constacyclic codes over the finite commutative non-chain ring R = Zp +νZp where ν2 = ν and Zp is field having p elements with characteristic p where p is prime. A Gray map is defined between R and Zp. The parameters of quantum codes over Zp are obtained by decomposing (1+(p−2)ν)-constacyclic codes into cyclic and negacyclic codes over Zp. As an application, some examples of quantum codes of arbitrary length, are also obtained.


INTRODUCTION
There has been an enormous development in the research on quantum codes. As the disclosure that quantum codes secure quantum information similar to classical codes classic information. Quantum information can propagate faster than light under certain conditions, while classical information cannot. Quantum information can't be duplicated but classical information can be. Quantum codes provide the most efficient way to overcome decoherence. The first quantum code was found by Shor [7]. From then on, the construction of quantum codes through classical cyclic codes and their generalizations haRs developed rapidly. Quantum codes attracted world wide attention therefore. Later on, Calderbank et al. [1] gave a technique to build quantum codes through classical codes in 1998. Recently the theory of quantum codes is on the path of everlasting development. In recent years, the theory of quantum code has been developed rapidly (see reference [4,8,9]). A significant development in the construction of quantum codes through cyclic codes over finite chain ring F 2 + uF 2 where u 2 = 0 of odd length was made by Qian [2]. Kai and Zhu [10] also gave a method to construct quantum codes through cyclic codes over finite chain ring F 4 + uF 4 where u 2 = 0 of odd length. Qian [3] studied quantum codes of arbitrary length through cyclic codes over finite non-chain ring F 2 +vF 2 where v 2 = v. Recently, Ashraf and Mohammad [5] defined the construction of quantum codes through cyclic codes over finite non-chain ring F 3 +vF 3 where v 2 = 1. Then in [6] Ashraf and Mohammad studied this topic over the different finite nonchain ring F q + vF q where v 2 = v. In this paper, encouraged by these type of problems, we study quantum codes through (1 + (p − 2)ν)-constacyclic codes over finite non-chain ring Z p + νZ p where ν 2 = ν. This paper is structured as follows. Section 2 contains preliminaries that deal with some basic properties of the considered ring and some basic definitions. In section 3, Gray Map is defined over the considered ring and the construction of quantum codes through constacyclic codes over the considered ring are given. Some examples are provided to illustrate the main result in section 4. Finally, paper is concluded in section 5.
The hamming weight w H (χ) for any codeword χ = (χ 0 , χ 1 , ..., χ n−1 ) ∈ ℜ n is defined as the number of all non-zero components in χ = (χ 0 , χ 1 , ..., χ n−1 ). The minimum weight of a code K , that is, w H (K ) is the least weight among all of its non zero codewords. The Hamming distance between two codes χ = (χ 0 , χ 1 , ..., χ n−1 ) and ψ = (ψ 0 , ψ 1 , ..., ψ n−1 ) of ℜ n , denoted by d H (χ, ψ) = w H (χ − ψ) and is defined as Minimum distance of K , denoted by d H and is given by minimum distance between the different pairs of codewords of the linear code K . For any codeword χ = (χ 0 , χ 1 , ..., χ n−1 ) ∈ ℜ n , the lee weight is defined as w L (χ) = ∑ n−1 i=0 w L (χ i ) and lee distance of (χ,χ) is given by Minimum lee distance of K is denoted by d L and is given by minimum lee distance of different pairs of codewords of the linear code K .

QUANTUM CODES OBTAINED THROUGH
This map can be extended to ℜ n , that is ϕ: It is obvious that, the map ϕ is linear and distance preserving isometry from (ℜ n , d L ) to (Z 2n p , d H ), where d L and d H are the lee distance and hamming distance in ℜ n and Z 2n p respectively. For a linear code K of length n over ℜ, we characterize Therefore, K ∞ and K ∈ are linear codes over the ring Z p of length n. Moreover, the linear code K can be uniquely expressed as and also |K |= |K ∞ ||K ∈ |.
The following proposition can be obtained directly by the above defined Gray map ϕ.
Proposition 3.1. Let K be a linear code over the ring ℜ of length n. If K is self orthogonal, then ϕ(K ) is also self orthogonal.
First we assume that K is a (1 + (p − 2)ν)-constacyclic code over the ring ℜ of length n then, which is an element of the linear code K . Therefore, K ∞ is a cyclic and K ∈ is a negacyclic codes over the ring Z p of length n.
The following lemma is similar to Theorem 4.2 [11].
Lemma 3.4. Let K be a (1 + (p − 2)ν)-constacyclic codes over the ring ℜ of length n. Then where g i (x) for i = 1, 2 are the generator polynomials of K ∞ and K ∈ respectively.
ν)-constacyclic codes over the ring ℜ of length n and where g i (x) for i = 1, 2 are reciprocal polynomials of x n +1 g 1 (x) and x n −1 g 2 (x) respectively.
First we consider x n − 1 ≡ 0 mod(g 1 (x)g 1 (x)) for K ∞ and x n + 1 ≡ 0 mod(g 2 (x)g 2 (x)) for K ∈ respectively, then by above lemma, we have Conversely, let us consider K ⊥ ⊆ K , then which implies that therefore we have By the above Theorem, we have the following corollary. The following theorem defines the construction of quantum codes by the use of Corollary 3.7 and Lemma 3.8.
constacyclic codes over the ring ℜ of length n where g i (x) are generator polynomials of K ∞ and K ∈ for i = 1, 2 respectively. If K ⊥ ∞ ⊆ K ∞ and K ⊥ ∈ ⊆ K ∈ then K ⊥ ⊆ K and there exists a quantum code having parameters [2n, 2k − 2n, ≥ d L ] p where k is the dimension of linear code ϕ(K ) and d L is minimum Lee distance of K .

EXAMPLES
In this section, some examples are provided to illustrate the main result. Here, the quantum codes through (1 + (p − 2)ν)-constacyclic codes over the ring ℜ = Z p + νZ p where ν 2 = ν are also obtained.

CONCLUSION
In this work, we have given a construction for quantum codes through (1 + (p − 2)ν)constacyclic codes over the finite non-chain ring ℜ = Z p + νZ p where ν 2 = ν. We have derived self-orthogonal codes over the ring Z p as Gray images of linear codes over the ring Z p + νZ p .
In particular, the parameters of quantum codes over the ring Z p are obtained by decomposing (1 + (p − 2)ν)-constacyclic codes into cyclic and negacyclic codes over the ring Z p .