QUANTUM CODES OVER A CLASS OF FINITE ... - Semantic Scholar

Quantum Information and Computation, Vol. 16, No. 1&2 (2016) 0039–0049 c Rinton Press

QUANTUM CODES OVER A CLASS OF FINITE CHAIN RINGS

MUSTAFA SARIa Department of Mathematics, Faculty of Art and Sciences, Yildiz Technical University Istanbul, 34210, TURKEY IRFAN SIAPb Department of Mathematics, Faculty of Art and Sciences, Yildiz Technical University Istanbul, 34210, TURKEY

Received November 4, 2014 Revised September 4, 2015 In this study, we introduce a new Gray map which preserves the orthogonality from the chain ring F2 [u] / (us ) to F2s where F2 is the finite field with two elements. We also give a condition of the existence for cyclic codes of odd length containing its dual over the ring F2 [u] / (us ) . By taking advantage of this Gray map and the structure of the ring, we obtain two classes of binary quantum error correcting (QEC) codes and we finally illustrate our results by presenting some examples with good parameters. Keywords: Quantum codes, Cylic codes, Gray map. Communicated by: S Braunstein & K Moelmer

1

Introduction

Quantum error-correcting (QEC) codes besides offering error correction in quantum communication also find applications in quantum computations. However, one and the most important problem in quantum computation is decoherence of quantum bits. To overcome this problem quantum error-correcting codes are first introduced by Shor [1]. In this first example a quantum error correcting code that can store the information of one qubit into a highly entangled state of nine qubits is presented. Later, Calderbank et al. gave a systematic way of constructing quantum error-correcting codes from classical error-correcting codes [2]. This method is known as the CSS construction method. After this novel approach introduced by Shor and Calderbank, the studies on this direction have increased significantly. We list some recent and most relevant studies for the convenience of the readers but surely we can not cover them all here. The CSS constructions that are going to be presented in the next section are based on classical error correcting codes and their relations with orthogonality. In [3], BCH codes containing their Euclidean and Hermitian duals which enable construction of quantum codes are studied. Constructing new QEC or families of such codes is another direction taken by the researchers. Recently, an open problem posed by Calderbank et al. in [2], has been settled by Tonchev in [4]. The studies on QEC as the original one has been mostly [email protected] b [email protected]

39

40

Quantum Codes

on binary codes but later these studies has been generalized to nonbinary codes with the goal to relate the later codes to the former ones. For instance, in [5], as a generalization of [2], nonbinary error bases are defined on Fq2 and for any prime power q, q-ary quantum codes are obtained from self-orthogonal codes over Fq2 . Further in [6], it is shown that there is a relation between quantum error correcting codes and classical linear codes and also a construction for quantum error correcting codes from linear codes over finite field is given. In [2], the problem of finding quantum-error-correcting codes is transformed into the problem of finding additive codes over the field F4 which are self-orthogonal with respect to a certain trace inner product. Recently, codes over rings that serve as a source for QEC have also been of interest. In [7], quantum codes are obtained from the cyclic codes over the ring R = F2 + vF2 + v 2 F2 and in [8], the structures of cyclic codes over the ring F2 [u]/(uk ) and their duals are studied. In the thesis [9], a new construction for quantum codes is given. Quantum codes are defined to be as the quantum state space fixed by a subgroup of the group consisting of all quantum errors. In [10], a new but simple construction of stabilizer codes is proposed based on syndrome assignment by classical parity-check matrices. In [11], the structures of linear and cyclic codes over a finite chain ring are studied. In [12], quantum codes from cyclic codes over the finite ring of four elements F2 + vF2 are constructed. In [13], by using insights provided by the classical theory of error correcting codes, the relation between quantum error correction and the theory of classical linear codes are studied. Also, some methods of error correction in the quantum regime are presented. In [14], a generalization of CSS construction is given, which enables many new codes of good efficiency to be discovered. And finally but surely not all, in [15], a preserving-orthogonality Gray map is defined from F2 + uF2 + u2 F2 to F23 and by making use of this Gray map quantum codes are obtained from cyclic codes containing their duals over this ring. The presentation of this paper is as follows: in section 2, we give basic definitions which are needed in the rest of this paper. In section 3, we define a Gray map preserving the orthogonality from F2 [u] / (us ) to F2s and define Lee weight and metric over the ring F2 [u] / (us ). In section 4, we give a condition for cyclic codes with odd length containing their duals over the ring F2 [u] / (us ) and obtain two classes of binary quantum error correcting codes as the Gray images of cyclic codes of odd length over the ring F2 [u] / (us ) which are containing their duals. In section 5, we illustrate the findings in this study by representing some selected examples with good parameters. Finally, section 6 concludes the paper. 2

Preliminaries

Let Fq be a finite field with q elements where q is a prime power. A code C over Fq of length n is a nonempty subset of Fqn and a linear code C over Fq of length n is a nonzero subspace of Fqn . The number |C| is called the size of the code C. Since a linear code C is a subspace, the code C has a dimension, denoted by k. An element of the code C is called a codeword. The Hamming weight wH (x) of a vector x ∈ Fqn is the number of nonzero coordinates of x. The Hamming weight wH (C) of a code C is the minimum nonzero weight in the code C. The Hamming distance dH (x, y) between two vectors x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ) is defined to be dH (x, y) = wH (x − y) .

(1)

M. Sari and I. Siap

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Also, the Hamming distance of a code C is defined by dH (C) = min {d (x, y) : x, y ∈ C, x 6= y} . In the case when the code is a linear code then its Hamming weight and distance are equal and this dramatically reduces the number of computations while one tries to determine this parameter which determines the correction capability of the code. A code C over Fq of the length n, the size M and the Hamming distance dH is denoted by (n, M, dH )q . Moreover, a linear code C over Fq of the length n, the dimension k and the Hamming distance dH is denoted by [n, k, dH ]q . Let C be a linear code over Fq of length n. The dual code C ⊥ of the code C with respect to usual inner product is the set  C ⊥ = y ∈ Fqn : hx, yi = 0, ∀x ∈ C .

It is well known that if C is a linear code with the length n and the dimension k, then C ⊥ is a linear code with the length n and the dimension n − k. A significant application of linear codes over Fq with some orthogonality conditions is used for constructing quantum error correcting codes. In this paper, since we present a Calderbank-Shor-Steane (CSS) construction we explain briefly how one can construct a QEC code by using classical error correcting codes. For a given linear code C1 of length n and dimension k1 , let C2 ⊂ C1 be a subcode of dimension k2 . For each additive coset Di = {vi + C2 |vi ∈ C1 } (1 ≤ i ≤ |C1 |/|C2 |) of C2 in C1 , we define a quantum codeword vˆ = p

1

X

|C2 | d∈Di

|di.

Then, the quantum code obtained from linear codes C1 and C2 is defined to be the vector space spanned by all vˆ. Here, the code C1 is used for bit-flip error correction, and the code C2⊥ is used for phase-flip by means of a Hadamard transform. For further details, the readers may refer to some basic and recent studies in [16, 17]. A quantum code over Fq with length n, dimension k and minimum distance d is denoted by [[n, k, d]]q . We state the following important theorem [6, 13]. Theorem 1:[6, 13](CSS Construction) Let C1 and C2 be two the linear codes with parameters [n, k1 , d1 ]q and [n, k2 , d2 ]q such that C2 ⊆ C1 , respectively. exists a quantum error   Then, there where d⊥ correcting code over Fq with parameters n, k1 − k2 , min d1 , d⊥ 2 is the Hamming 2 q ⊥ weight of the code C2 . Then, one gets the following easily: Corollary 2: Let C be a linear code having the parameters [n, k, dH ]q such that C ⊥ ⊆ C. Then, there exists a quantum error correcting code having the parameters [[n, 2k − n, dH ]]q . Example 3: It is well-known that the binary Hamming code with parameters [7, 4, 3] contains its dual. By using the CSS construction, we have a [[7, 1, 3]] quantum error correcting code.

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Quantum Codes

We also have the following from [14]. Theorem 4:[14](Steane’s Construction) Let C and C1 be two binary linear codes with the parameters [n, k, d] and [n, k1 , d1 ] such that C ⊥ ⊂ C ⊂C1 and k1 ≥ k + 2. Then,   exists  there . a quantum error correcting code with the parameters n, k + k1 − n, min d, 23 d1 Hence, the study of finding linear codes containing their duals is very important since it not only provides quantum codes but also gives the parameters of quantum error correcting codes obtained via CSS construction or Steane’s construction. Let R be a commutative ring with identity 1 6= 0. If the ideals of the ring R form a finite chain with respect to set inclusion, then R is called a finite chain ring. Let γ be a generator of maximal ideal of the ring R. It is well known that the ideals of the ring R have the following form:
{0} = (γ e ) ⊂ γ e−1 ⊂ · · · ⊂ (γ) ⊂ R

where e is the smallest positive integer such that γ e = 0, which is called the nilpotency index of the ring R. The quotient ring R/ (γ) is called the residue field of the ring R having q elements where q is a prime power. We have the following from [11].

Theorem 5:[11] Let V be a maximal subset of R and γ be a generator of maximal ideal of R with the property that x1 6= x2 mod (γ) for all x1 , x2 ∈ R such that x1 6= x2 . Then, 1. for all x ∈ R, there are unique x0 , x1 , . . . , xe−1 ∈ V such that x = x0 + x1 γ + · · · + xe−1 γ e−1 ; e−j 2. γ j R = |Fq | for 0 ≤ j ≤ e. Example 6: The quotient ring F2 [u] / (us ) is a finite chain ring with the maximal ideal (u), the residue field F2 and the nilpotency index s. Every element x in F2 [u] / (us ) has the unique form x = x0 + x1 u + · · · + xs−1 us−1 where xi ∈ F2 for 0 ≤ i ≤ s − 1. Throughout the rest of this paper we denote R = F2 [u] / (us ). 3

A Gray Map from Rn to F2sn


In [15], Xunru and Wenping defined an orthogonality preserving Gray map from F2 [u] / u3 to F23 . In this section, we are going to extend this definition from Rn to F2sn . We associate a map from R to F2s with an s × s matrix such that the ith row of the matrix represents the ith component of the image and the ith column of the matrix represents ai−1 in the components of the image for an element a0 + a1 u + · · · + as−1 us−1 ∈ R. In order to obtain a binary code from the codes over the ring R, we introduce first an auxiliary map defined through a special matrix as follows: Definition 7: Let A be an s × s square  0 (mod2) , T 1. If s is odd, Ai Aj = 1 (mod2) ,  0 (mod2) , 2. If s is even, ATi Aj = 1 (mod2) ,

matrix such that if i + j = 6 s+1 , if i + j = s + 1 if i + j = 6 s+1 but AT1 A1 6= 0 (mod2), if i + j = s + 1

M. Sari and I. Siap

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where Ai is the ith column of the matrix A. We define the map ϕ : R → F2s by ϕ (x) = (a0 , a1 , . . . , as−1 )AT where x = a0 + a1 u + · · · + as−1 us−1 ∈ R. In this case, we denote such a map by A ϕ. Note thatthe matrix A defined above for a fixed value s. For instance, for  is not unique  0 1 1 1 1 0 s = 3, A1 =  1 1 1  and A2 =  1 1 1  hold the condition (1) given in Definition 1 1 0 0 1 1 7 but A1 6= A2 . Corollary 8: The matrix A defined in the Definition 7 is invertible. Proof: Observe that for odd s 

0  ..  . AT A =    0 1 and for even s



     AT A =     

1 0 0 .. . .. . 0 1

0 .. . 0 1 0

··· .. . . .. 0 ··· .. . .. . . .. 0 ···

0 . .. ..

. ···

1 0 .. . 0

     

···

0

1

0 . ..

1

0 .. . .. . 0 0

..

. ··· ···

0 .. 0 0

.

Since AT A has full rank in both cases, A is invertible.



     .    

Theorem 9: The matrix A defined in the Definition 7 always exists.

Proof: First, we give a construction for the matrix A when s is odd. Let A3×3 and let s be odd. For all s ≥ 3, define recursively   B2×s |~y A(s+2)×(s+2) = ~x| As×s 



0 1 = 1 1 1 1

 1 1  0

1, if j = s+1 2 , 0, otherwise. Next, we give a construction for the matrix A when s is even. Let s be even and Aj be j th column of the matrix A. Define T

T

where ~x = (011 · · · 1)1×s , ~y = (11 · · · 10)1×s and B2×s = (bij ) =

AT1 = (00 . . . 01)1×s , ATs = (11 . . . 1)1×s ,  

s   f or 2 ≤ j ≤ , . . . 1} 00 . . . 0 ATj =  11 | {z 2 2.( 2s −j+1) 1×s

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Quantum Codes

and



 ATj = 



s  00 . . . 0} 1100 . . . 0 f or < j < s − 1. | {z 2 2.(j− 2s −1)+1 1×s

Then, it is easy to see that the matrix A holds the condition (1) and (2) for the odd and even values of s, respectively. Definition 10: Let the s×s matrix A and the map A ϕ be defined as above. Let (x1 , x2 , . . . , xn ) ∈ Rn . We define a Gray map ϑ from Rn to F2sn by ϑ:

(x1 , x2 , . . . , xn ) → (A ϕ (x1 ) , A ϕ (x2 ) , . . . , A ϕ (xn )) .

Theorem 11: The Gray map ϑ preserves the orthogonality from Rn to F2sn . Proof: Let x and y be two elements in R such that x⊥y. By polynomial product,

P

x i yj =

i+j=k

0 for all 0 ≤ k ≤ s − 1. Observe that the number of xi yj appearing in hϑ (x) , ϑ (y)i is exactly ATi+1 Aj+1 . Then we get the followings: P xi yj = 0. 1. If s is odd, hϑ (x) , ϑ (y)i = i+j=s−1

2. If s is even, hϑ (x) , ϑ (y)i = x0 y0 +

P

xi yj = 0.

i+j=s−1

Hence, proof is complete. Since the Gray map ϑ is a linear map and by Theorem 11, we have: Theorem 12: If C is a linear code containing its dual over R, then ϑ (C) is so. 

 0 0 1 0 1  1 0 1 0 1     Example 13: Let A =   1 0 1 1 0 . Since the matrix A holds the definition, the  1 1 1 1 0  1 1 1 0 0
associated Gray map ϑ is a map which preserves the orthogonality
from F2 [u] / u5 to F25 and carries an element a0 + a1 u + a2 u2 + a3 u3 + a4 u4 in F2 [u] / u5 to (a2 + a4 , a0 + a2 + a4 , a0 + a2 + a3 , a0 + a1 + a2 + a3 , a0 + a1 + a2 ) .

The Lee weight wL (x) of an element x ∈ R is defined as wL (x) = wH (ϑ (x)). The Lee weight is then extended to Rn componentwise i.e. if x = (x1 , x2 , . . . , xn ) ∈ Rn , then its Lee weight is n X wL (xi ). wL (x) = i=1

The Lee distance dL (x, y) between x = (x1 , x2 , . . . , xn ) and y = (y1 , y2 , . . . , yn ) is defined as dL (x, y) = wL (x − y) .

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Then we have the followings: Theorem 14: The Gray map ϑ is a distance preserving map from (Rn , dL ) to (F2sn , dH ). Corollary 15: Let K be the size of the code C over R. If C is an (n, K, dL ) linear code, then ϑ (C) is an (sn, K, dL ) linear code over F2 . 4

Quantum Codes from Cyclic Codes over R of Odd Length

A code C is called cyclic over R of length n if (cn−1 , c0 , c1, . . . , cn−2 ) ∈ C for every codeword (c0 , c1 , . . . , cn−1 ) ∈ C. It is well known that there is a one-to-one correspondence between the cyclic codes over R of length n and the ideals in the quotient ring R [x] / (xn − 1). The ideal structure of R [x] / (xn − 1) for odd values of n was completely determined in [8]. Theorem 16:[8] Suppose that C is a cyclic code over R of odd length n, then there exits monic polynomials f0 , f1 , . . ., fs such that
1. C = G1 , uG2 , . . . , us−1 Gs where Gi = where t =

s P

xn −1 fi

for 1 ≤ i ≤ s, xn − 1 =

s Q

fi and |C| = 2t

i=0

(s − i) deg (Gi+1 ).

i=0


R s−1 R 2. C ⊥ = GR G2 where Gi = 0 , uGs , . . . , u s Q

i=0


deg(Gi ) fi and GR Gi x−1 . i =x

xn −1 fi

for 2 ≤ i ≤ s, G0 =

s Q

i=1

xn −1 Gi ,

xn − 1 =

We now give a condition for a cyclic code of odd length over R containing its dual. Theorem 17: Suppose that C is a cyclic code of odd length n over the ring R and C ⊥ = s
Q n R s−1 R . Then, C ⊥ ⊆ C if and only if fi and Gi = x f−1 G2 where xn − 1 = GR 0 , uGs , . . . , u i

for all 1 ≤ i, j ≤ s − 1, xn − 1 divides

i=0

1. ui+j GR s−i+1 Gs−j+1 , 2. uj GR 0 Gs−i+1 and 3. GR 0 G0 . By combining Theorem 17, Theorem 12 and Corollary 2, we have the following: Theorem 18: Suppose that C is a cyclic code of odd length n over the ring R and C = s
Q n fi and Gi = x f−1 G1 , uG2 , . . . , us−1 Gs where xn − 1 = . If for all 1 ≤ i, j ≤ s − 1, xn − 1 i i=0

divides

1. ui+j GR s−i+1 Gs−j+1 , 2. uj GR 0 Gs−i+1 and 3. GR 0 G0 ,

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Quantum Codes

then there exists a family of binary quantum error correcting codes with the parameters s−1 P (s − i) deg (Gi+1 ). [[ns, 2t − n, dL ]] where dL is the minimum Lee distance of the code C and t = i=0

By combining Theorem 17, Theorem 12 and Theorem 4, we also have the following:

Theorem 19: Suppose that C is a cyclic code of odd length n over R and C = G1 , uG2 , . . . , us−1 Gs s Q n fi and Gi = x f−1 where xn − 1 = . Let C1 be a code containing the cyclic code C and the i




i=0

gray image ϑ (C1 ) has the dimension k1 and the minimum distance d1 . If for all 1 ≤ i, j ≤ s−1, xn − 1 divides 1. ui+j GR s−i+1 Gs−j+1 , 2. uj GR 0 Gs−i+1 and 3. GR 0 G0 ,

then there exists a family binary quantum error correcting codes with the parameters   of    ns, t + k1 − n, min dL , 23 d1 where dL is the minimum Lee distance of the code C and s−1 P (s − i) deg (Gi+1 ). t= i=0

5

Examples

In this section we present some examples that also are tabulated in order to illustrate the findings presented in the previous sections. We note that xn − 1 has the same factorization over both F2 and R since the characteristics of F2 and R are the same.

Example 20: Let x7 −1 = (1 + x) 1 + x2 + x3 1 + x + x3 = G1 G2 , where G1 = 1+x+x3 and G2 = 1+x+x2 +x4 . Some of cyclic codes over R of length 7 and quantum codes obtained via Theorem 18 and Theorem 19 are given in Table 1 and Table 2, respectively. Table 1. Some of the binary quantum codes obtained via Theorem 18 in Example 20. R s=2 s=2 s=3 s=3 s=4 s=4 s=4 s=5 s=5 s=5

C (G1 ) (G1 , uG2 ) (G1 ) (G1 , uG2 ) (G1 )
G 1 , u2 G 2 (G1 , uG2 ) (G1 )
G 1 , u2 G 2 (G1 , uG2 )

ϑ (C) [14, 8, 3] [14, 11, 2] [21, 12, 3] [21, 18, 1] [28, 16, 3] [28, 22, 2] [28, 25, 2] [35, 20, 3] [35, 29, 1] [35, 32, 1]

Quantum Code [[14, 2, 3]] [[14, 8, 2]] [[21, 3, 3]] [[21, 15, 1]] [[28, 4, 3]] [[28, 16, 2]] [[28, 22, 2]]∗ [[35, 5, 3]] [[35, 23, 1]] [[35, 29, 1]]

Matrix A1 A1 A2 A2 A3 A3 A3 A4 A4 A4

Remark: The parameter with * in the Table 1 is optimal by Markus Grassl’s table (See http://www.codetables.de/).



Example 21: Let x15 −1 = (1 + x) 1 + x + x2 1 + x + x4 1 + x3 + x4 1 + x + x2 + x3 + x4 = G1 G2 , where G1 = 1 + x3 + x4 and G2 = 1 + x3 + x4 + x6 + x8 + x9 + x10 + x11 . Some of

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Table 2. Some of the binary quantum codes obtained via Theorem 19 in Example 20. R s=2 s=3 s=4 s=4 s=4 s=5 s=5 s=5

C (G1 ) (G1 ) (G1 ) (G1 )
G 1 , u2 G 2

(G1 ) (G1 )
G 1 , u2 G 2

C1 (G1 , uG2 ) (G1 , uG2 )
G 1 , u2 G 2 (G1 , uG2 ) (G1 , uG2 )

Quantum Code [[14, 5, 3]] [[21, 9, 2]] [[28, 10, 3]] [[28, 13, 3]] [[28, 19, 2]]

Matrix A1 A2 A3 A3 A3

G 1 , u2 G 2 (G1 , uG2 ) (G1 , uG2 )

[[35, 14, 2]] [[35, 17, 2]] [[35, 26, 1]]

A4 A4 A4




cyclic codes over R of length 15 and quantum codes obtained via Theorem 18 and Theorem 19 are given in Table 3 and Table 4, respectively. Table 3. Some of the binary quantum codes obtained via Theorem 18 in Example 21. R s=2 s=2 s=3 s=3 s=4 s=4 s=4 s=5 s=5 s=5

C (G1 ) (G1 , uG2 ) (G1 ) (G1 , uG2 ) (G1 )
G 1 , u2 G 2 (G1 , uG2 ) (G1 )
G 1 , u2 G 2 (G1 , uG2 )

ϑ (C) [30, 22, 3] [30, 26, 2] [45, 33, 3] [45, 41, 1] [60, 44, 3] [60, 52, 2] [60, 56, 2] [75, 55, 3] [75, 67, 1] [75, 71, 1]

Quantum Code [[30, 14, 3]] [[30, 22, 2]] [[45, 21, 3]] [[45, 37, 1]] [[60, 28, 3]] [[60, 44, 2]] [[60, 52, 2]] [[75, 35, 3]] [[75, 59, 1]] [[75, 67, 1]]

Matrix A1 A1 A2 A2 A3 A3 A3 A4 A4 A4

Table 4. Some of the binary quantum codes obtained via Theorem 19 in Example 21. R s=2 s=3 s=4 s=4 s=4 s=5 s=5 s=5

C (G1 ) (G1 ) (G1 ) (G1 )
G 1 , u2 G 2

(G1 ) (G1 )
G 1 , u2 G 2

C1 (G1 , uG2 ) (G1 , uG2 )
G 1 , u2 G 2 (G1 , uG2 ) (G1 , uG2 )

Quantum Code [[30, 18, 3]] [[45, 29, 2]] [[60, 36, 3]] [[60, 40, 3]] [[60, 48, 2]]

Matrix A1 A2 A3 A3 A3

G 1 , u2 G 2 (G1 , uG2 ) (G1 , uG2 )

[[75, 47, 2]] [[75, 51, 2]] [[75, 63, 1]]

A4 A4 A4






Example 22: Let x21 −1 = 1 + x + x2 + x4 + x6 1 + x + x3 + x6 + x7 + x10 + x13 + x15 = G1 G2 . Some of cyclic codes over R of length 21 and quantum codes obtained via Theorem 18 and Theorem 19 are given in Table 5 and Table 6, respectively. The matrices A1 , A2 , A3 and A4 given in the Tables 1, 2, 3, 4, 5 and 6 are as follows:     0 0 1 0 1   1 1 0 1  1 0 1 0 1    0 1 1    1 1 1 1  0 1     1 0 1 1 0  1 1 1 , A4 =  A1 = , A3 =  , A2 = .   1 1 1 0 1 1  1 1 1 1 0  1 1 0 0 0 0 1 1 1 1 0 0

48

Quantum Codes Table 5. Some of the binary quantum codes obtained via Theorem 18 in Example 22. R s=2 s=2 s=3 s=3 s=4 s=4 s=4 s=5 s=5 s=5

C (G1 ) (G1 , uG2 ) (G1 ) (G1 , uG2 ) (G1 )
G 1 , u2 G 2 (G1 , uG2 ) (G1 )
G 1 , u2 G 2 (G1 , uG2 )

ϑ (C) [42, 30, 3] [42, 36, 2] [63, 45, 3] [63, 57, 1] [84, 60, 3] [84, 72, 2] [84, 78, 2] [105, 75, 3] [105, 93, 1] [105, 99, 1]

Quantum Code [[42, 18, 3]] [[42, 30, 2]] [[63, 27, 3]] [[63, 51, 1]] [[84, 36, 3]] [[84, 60, 2]] [[84, 72, 2]] [[105, 45, 3]] [[105, 81, 1]] [[105, 93, 1]]

Matrix A1 A1 A2 A2 A3 A3 A3 A4 A4 A4

Table 6. Some of the binary quantum codes obtained via Theorem 19 in Example 22. R s=2 s=3 s=4 s=4 s=4 s=5 s=5 s=5

6

C (G1 ) (G1 ) (G1 ) (G1 )
G 1 , u2 G 2 (G1 ) (G1 )
G 1 , u2 G 2

C1 (G1 , uG2 ) (G1 , uG2 )
G 1 , u2 G 2 (G1 , uG2 ) (G1 , uG2 )
G 1 , u2 G 2 (G1 , uG2 ) (G1 , uG2 )

Quantum Code [[42, 24, 3]] [[63, 39, 2]] [[84, 48, 3]] [[84, 54, 3]] [[84, 66, 2]] [[105, 63, 2]] [[105, 69, 2]] [[105, 87, 1]]

Matrix A1 A2 A3 A3 A3 A4 A4 A4

Conclusion

We generalize the results given in [15] to a class of finite chain ring R = F2 [u] / (us ). By associating with an s × s square matrix, we define a Gray map ϑ from Rn to F2sn that preserves the orthogonality and we study its properties. We further obtain a condition for a cyclic code over R of odd length that contains its dual. By making use of the Gray images of such codes, we construct two families of binary quantum error correcting codes. Finally, we give some examples of binary quantum error correcting codes derived in section 4. The advantage of this approach is that the orthogonal codes over the ring R are easier to represent and more compact comparing to the direct constructions from binary fields. Obtaining QEC from linear codes over R of even length is an open and interesting problem for future studying. Also, using the same approach as presented in this paper, QEC codes obtained from different rings can also be considered for future work. Acknowledgements This research is partially supported by the grant of Yildiz Technical University Research Support Unit, Project No: 2013-01-03-KAP02. The authors also would like to thank the anonymous referees for their time and valuable remarks. References 1. P. W. Shor, (1995), Scheme for reducing decoherence in quantum memory, Phys. Rev. A, Vol. 52, pp. 2493-2496. 2. A. R. Calderbank, E. M. Rains, P. M. Shor, N. J. A. Sloane, (1998), Quantum error correction

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