QUANTUM CODES FROM CYCLIC CODES OVER F₄ + vF₄

Abstract

In this work, a method is given to construct quantum codes from cyclic codes over F₄ + vF₄ which will be denoted as R throughout the paper, where v² = v and a Gray map is defined between R and where F₄ is the field with 4 elements. Some optimal quantum code parameters and others will be presented at the end of the paper.

1. Introduction

Error correcting quantum codes are being used to deal with quantum noises such as decoherence which arises after by inventing quantum computers. The first discovery of error correcting quantum codes is made by Shor in [1]. After that discovery,a method is given for constructing quantum codes from widely known classical error correcting codes by Calderbank et al. in [2]. Recently, another widely known class of error correcting codes, cyclic codes over the field are being used in the purpose of obtaining quantum error correcting codes, where q is a power of prime. In [6], a technique is given to construct for quantum error correcting codes over the finite ring by Qian. Kai and Zhu in [7] gave a technique to construct quantum error correcting codes from cyclic codes with length n , where n is an odd integer, over finite chain ring with u2 = 0. Moreover, Qian [5] gave an original method which uses cyclic codes over the finite ring with v2 = v, to construct quantum error correcting codes. Motivated by this study, M.Ashraf [8] describes a similar construction method for quantum codes which is obtained from cyclic codes over where v2 = 1. Making use of the cyclic codes over finite ring where v2 = v, we obtain quantum codes over The structure of cyclic codes over was given by A. Bayram in [9]. It was shown that this ring is isomorphic to

In this paper, a method is described for obtaining self-orthogonal codes over as Gray map images of linear and cyclic codes over the ring v2 = v. A sufficient and necessary condition for cyclic codes over R which contains its dual is given. At the end, parameters of associated quantum codes will be presented, some of them are optimal based on the table in [10].

 

2. Preliminaries

Let where v2 = v, where w2 = w + 1. R is a finite non-chain ring with 16 elements. The ring R has two maximal ideals (v) = {0, v, vw, v(w + 1)} and (1 + v) = {0, v + 1, (v + 1)w, (v + 1)(w + 1)}. If we apply the Chinese Remainder Theorem, it is obtained that which means that, every element of R can be expressed uniquely as x + vy = v(x + y) + (v + 1)x, for some If γ = (γ1, γ2, . . . , γn) ∈ Rn, then the Hamming weigh of γ is the number of nonzero coordinates in γ and is represented by hw(γ). d(γ, δ) = hw(γ − δ) gives the Hamming distance between γ and δ in R.

A linear code C over R of length n is an R submodule of Rn. If γ ∈ C, then we say that γ is a codeword. C is a cyclic code, if γ = (γ0, γ1, . . . , γn−1) ∈ C, then (γn−1, γ0, γ1, . . . , γn−2) ∈ C. Let γ = (γ1, γ2, . . . , γn) and δ = (δ1, δ2, . . . , δn) be two elements of Rn. Then the Euclidean inner product of γ and δ in Rn is defined as

If C is a linear code of length n over R, then its dual code is defined as

We say that a code C is a self-orthogonal code, if it satisfies the condition C ⊆ C⊥ and a self dual code if C = C⊥.

If A1 and A2 are two linear codes then their direct sum and cartesian product is denoted by A1⊗A2 = {(a1, a2)|a1 ∈ A1, a2 ∈ A2} and A1⊕A2 = {a1+a2|a1 ∈ A1, a2 ∈ A2} respectively.

It can be seen directly that by suitable permutation of coordinates, the generator matrix of a nonzero linear code C over R can be considered in the form:

where Ak and Bℓ are matrices over for k = 1, 2, 3, 4 and ℓ = 1, 2, 3, 4 . The next result is defined in [9]. Let

Clearly C1 and C2 are linear codes over . Thus, C = vC1 ⊕ (1 + v)C2 and |C| = 16k14k24k3.

Theorem 2.1 (CSS Construction). Let C has the parameter [n, k, d] and has the parameter then an quantum code can be obtained. In particular, if we take C⊥ ⊆ C, then a quantum code that has the parameters [n, 2k − n, d] can be obtained.

 

3. Gray Images of Linear Codes over

We know that every element of can be expressed as a + vb, where The Gray map ψ from R to is defined as ψ(a+vb) = (a+b, a). It is easy to show that ψ is a linear map. An extension of the Gray map ψ can be made in obvious way from Rn to

Proposition 3.1. The Gray map ψ is a map that preserves the distance from (Rn, Lee distance) to ( Hamming distance).

Proposition 3.2. Let C = vC1 ⊕ (1 + v)C2 be a linear code of length n over R and Ci be [n, ki, d(Ci)] linear codes for i = 1, 2. Then ψ(C) is a [2n, k1 + k2, min{d(C1), d(C2)}] code over

Proposition 3.3. Let C be a code of length n over R. Assume C is a self-orthogonal code. Then ψ(C) is self-orthogonal.

Proof. Let c1 = γ1 + vδ1 and c2 = γ2 + vδ2 ∈ C, where Then Euclidean inner product of c1 and c2, is

We know that C is self-orthogonal,then we have γ1γ2 = γ1δ2 + γ2δ1 + δ1δ2 = 0. On the other hand, ψ(c1)ψ(c2) = γ1γ2 + γ1δ2 + γ2δ1 + δ1δ2 + γ1γ2 = 0. Hence ψ(C) is self-orthogonal. □

 

4. Quantum Codes from Cyclic Codes over R

In this section, we use cyclic codes over R of arbitrary length n to obtain self-orthogonal codes over By using these self-orthogonal codes, the corresponding quantum code parameters will be determined.

The following lemmas are given in [4] and [9].

Lemma 4.1 ([9]). Assume that C = vC1 ⊕ (1 + v)C2 be a linear code over So C is a cyclic code over if and only if C1 and C2 are both cyclic codes over

Lemma 4.2 ([9]). If C = vC1 ⊕ (1 + v)C2 is a cyclic code of length n over Then C = ⟨vg1(x), (1 + v)g2(x)⟩ and |C| = 42n−deg(g1(x))−deg(g2(x)), where g1(x), g2(x) are the generator polynomials of C1,C2, respectively.

Lemma 4.3 ([4]). Let C be a cyclic code of length n over R. Then there exists unique polynomial g(x) such that C = ⟨g(x)⟩,and g(x)|xn − 1, where g(x) = vg1(x) + (1 + v)g2(x).

Lemma 4.4 ([4]). Let C = vC1 ⊕ (1 + v)C2 is a cyclic code of length n over R. Then and |C⊥| = 4deg(g1(x))+deg(g2(x)), where are the reciprocal polynomials of hi(x), that is, hi(x) = xn − 1/gi(x),

Now, we give a lemma that describes the necessary and sufficient condition for a cyclic code to be self-orthogonal.

Lemma 4.5. Let C be a cyclic code with generator polynomial g(x), then C contains its dual code if and only if

where g∗(x) is the reciprocal polynomial of g(x).

Now, a sufficient and necessary condition for cyclic code over R that contains its dual is given.

Theorem 4.6. Let C = ⟨g(x)⟩ is a cyclic code of length n over R, where g(x) = vg1(x) + (1 + v)g2(x). Then C⊥ ⊆ C if and only if

Proof. Let C = ⟨g(x)⟩ = vC1 ⊕ (1 + v)C2 be a cyclic code of length n over R, then C = ⟨vg1(x), (1 + v)g2(x)⟩, C1 = ⟨g1(x)⟩ and C2 = ⟨g2(x)⟩. If

Then

This implies that

Therefore

Hence

that is, C⊥ ⊆ C.

On the other hand if C⊥ ⊆ C, then Since C1 (resp.C2) be the code over such that vC1 (resp (1 + v)C2) is equal to C modv (resp. C mod(1 + v)), Therefore,

Corollary 4.7. Let C = vC1 ⊕ (1 + v)C2 be a cyclic code of length n over Then we have C⊥ ⊆ C if and only if

By Theorem 2.1 and Corollary 4.7, we can obtain quantum codes.

Theorem 4.8. Let C = vC1 ⊕ (1 + v)C2 is a cyclic code of arbitrary length n over R and let the parameters of ψ(C) be [2n, k, dL] where dL is defined as the minimum Lee weight of C. If

Then we have C⊥ ⊆ C and thus we get a quantum error-correcting code which has the parameters [[2n, 2k − 2n, dL]] where dL.

 

5. Examples

Example 5.1. Let and n = 10. Then

x10 −1 = (x+1)2(x2 +wx+1)2(x2 +w2x+1)2 in . Let g(x) = vg1(x)+(1+v)g2(x) with

and C =< g(x) > be a cyclic code over R. Clearly x10 − 1 is divisible by for i = 1, 2. Hence by Corollary 4.7 we have C⊥ ⊆ C. Then a quantum code with parameters [[20,16,2]] is obtained, which is optimal based on [10]. Other optimal codes are presented in Table 1.

TABLE 1.The Parameters of Optimal Quantum Codes

Example 5.2. Let and n = 36. Then

x36 − 1 = (x + 1)4(x + w)4(x + w2)4(x3 + w)4(x3 + w2)4 in . Let g(x) = vg1(x) + (1 + v)g2(x) with

and C =< g(x) > be a cyclic code over R. Clearly x36−1 is divisible by for i = 1, 2. Hence by Corollary 4.7 we have C⊥ ⊆ C. Then we obtain a quantum code with parameters [[72,34,4]]

Example 5.3. Let and n = 43. Then

x43 − 1 = (x + 1)(x7 + wx5 + x4 + x3 + w2x2 + 1)(x7 + w2x5 + x4 + x3 + wx2 + 1)(x7+x6+wx5+w2x2+x+1)(x7+x6+w2x5+wx2+x+1)(x7+wx6+wx5+ wx4 +w2x3+w2x2+w2x+1)(x7 +w2x6 +w2x5 +w2x4 +wx3 +wx2 +wx+1) in . Let g(x) = vg1(x) + (1 + v)g2(x) with

g1(x) = g2(x)

and C =< g(x) > be a cyclic code over R. Clearly x43−1 is divisible by for i = 1, 2. Hence by Corollary 4.7 we have C⊥ ⊆ C. Then a quantum code with parameters [[86,30,8]] is obtained.

Example 5.4. Let and n = 57. Then

x57 −1 = (x+1)(x+w)(x+w2)(x9 +x8 +wx6 +x5 +x4 +w2x3 +x+1)(x9 + x8 +w2x6 +x5 +x4 +x3 +x+1)(x9 +x8 +wx6 +wx5 +w2x4 +w2x3 +w2x+ 1)(x9 +wx8 +w2x6 +wx5 +w2x4 +wx3 +w2x+1)(x9 +w2x8 +wx6 +w2x5 + wx4 + w2x3 + wx + 1)(x9 + w2x8 + w2x6 + w2x5 + wx4 + wx3 + wx + 1) in Let g(x) = vg1(x) + (1 + v)g2(x) with

g1(x) = g2(x)

and C =< g(x) > be a cyclic code over R. Clearly x57−1 is divisible by for i = 1, 2. Hence by Corollary 4.7 we have C⊥ ⊆ C. Then a quantum code with parameters [[114,38,6]] is obtained.

TABLE 2.The Parameters of Quantum Codes