- Email: [email protected]
On Bounds for Quantum Error Correcting Codes over EJ-Integers
Available online at www.sciencedirect.com
Electronic Notes in Discrete Mathematics 70 (2018) 89–94
www.elsevier.com/locate/endm
On Bounds for Quantum Error Correcting Codes over EJ-Integers Eda YILDIZ 1 Department of Mathematics Yildiz Technical University Istanbul, Turkey
Abstract There are some differences between quantum and classical error corrections [4]. Hence, these differences should be considered when a new procedure is performed. In our recent study, we construct new quantum error correcting codes over different mathematical structures. The classical codes over Eisenstein-Jacobi(EJ) integers are mentioned in [3]. There is an efficient algorithm for the encoding and decoding procedures of these codes [3]. For coding over two-dimensional signal spaces like QAM signals, block codes over these integers p = 7, 13, 19, 31, 37, 43, 61, ... can be useful [2]. Thus, in this study, we introduce quantum error correcting codes over EJ-integers. This type of quantum codes may lead to codes with some new and good parameters. Keywords: Eisenstein-Jacobi integers, quantum codes, quantum error correcting codes, bounds on codes, QAM signals.
1
Email: [email protected]
https://doi.org/10.1016/j.endm.2018.11.015 1571-0653/© 2018 Elsevier B.V. All rights reserved.
90
1
E. Yildiz / Electronic Notes in Discrete Mathematics 70 (2018) 89–94
Introduction
Error correction is necessary during a data transmission. When a message is sent, some errors may occur in communication channel. Environmental effects, noises and the other interferences cause deformation on messages. Error correcting codes are used to correct these errors. While classical coding theory has been developing day by day, a new theory arised with an idea of a quantum computer. A new machine, which is based on quantum mechanics and called quantum computer, is expected to be more powerful than a classical computer. In 1976, Ingarden published a seminar paper and made a claim that classical information theory cannot be generalized directly to the quantum case. So, a new information theory based on quantum mechanics is needed. Hence, quantum error correction procedure is developed. The procedure is different and more complicated than the classical process. However, in 1996, a quantum code was invented by Robert Calderbank, Peter Shor and Andrew Steane. This code leads to a relation between classical linear codes and quantum codes. Thus, the problem of finding a quantum code transforms into finding a classical code containing the other code. This type of quantum code is known as CSS code. In general, there is no efficient decoding algorithm for block codes over two dimensional signal constellations such as QAM. So, different distances are used like Mannheim, Lee metric. It is better to develop decoding algorithm which works in an algebraic way for specific signal constellations. Huber, therefore, introduced linear codes over EJ-integers and defined a new two dimensional modular distance. He showed that there is an isomorphism between the set of EJ-integers with p element and finite field of characteristic p. The isomorphism leads to codes over these integers. Using of these codes may be more convenient for hexagonal signal constellations.
2
Preliminaries
In quantum channel, errors are represented by quantum operators. There are four important matrices X, Y, Z and I in quantum mechanics. They are called Pauli matrices and defined as follows: 01 0 −i 1 0 10 X = 1 0 , Y = i 0 , Z = 0 −1 , I = 0 1 In fact, any error can be represented as a linear combination of Pauli matrices. Since some features of these special matrices make it easier to calcu-
E. Yildiz / Electronic Notes in Discrete Mathematics 70 (2018) 89–94
91
late, they play an important role in the error correction procedure. Indeed, the set of Pauli matrices is not a group under the multiplication. However, it can be extended to a group, say P = {± X, ±iX, ±Y, ±iY, ± Z, ±iZ, ± I, ±iI }. Any Pauli operator on n qubits is a product of the elements of the group P. Moreover, a quantum code that encodes k qubits to n qubits is 2k -dimensional subspace of 2n -dimensional Hilbert space and it is denoted by [[n, k ]]. √
Definition 2.1 The set Z [ρ] = { a + bρ : a, b ∈ Z and ρ = −1+2 −3 } is a ring under the addition and multiplication. It is called ring of Eisentein-Jacobi integers and elements of the ring are called Eisenstein-Jacobi(EJ) integers. Primes of the form p ≡ 1 mod 6 can be written as sum of two integers such that p = a2 + 3b2 . Such primes are factored as ππ ∗ = p = a2 + 3b2 where π = a + b + 2b.ρ and π ∗ = a + b + 2b.ρ2 . There is a map μ : GF ( p) −→ Jπ defined by μ( a) = a −
aπ ∗ ππ ∗
π
where Jπ is the residue class of Z [ρ] modulo π and the operation [.] denotes rounding to the closest EJ integer. It is a one-to-one and onto homomorphism and hence, GF ( p) is isomorphic to Jπ . The isomorphism leads to a relation between a field of characteristic p and a set of EJ-integers with p elements. Here, each element of Jπ corresponds to a signal point. When each element of GF ( p) is represented as an element of Jπ , it provides important advantages for coding in the physical sense like hexagonal signal constellations. For the new codes, we also need to define a distance. The element α ∈ Jπ can be written in several different ways as α = g1 1 + g2 2 where 1 , 2 ∈ {±1, ±ρ, ±(1 + ρ)}. The weight of α is defined as w(α) = min{| g1 | + | g2 |} by Huber in [3]. The distance d between α1 and α2 is defined as d(α1 , α2 ) = w(α).
3
Quantum Codes over EJ Integers
Now, we will describe the structure of quantum codes over EJ-integers. Let p be a prime of the form p ≡ 1 mod 6. A p-ary quantum code C of length n and size k is a k-dimensional subspace of a pn -dimensional Hilbert space. We can define a new weight wt and a new distance d as follows: wt J (w) =
|+| g |}+...+ min {| g [min{| g0,1 |+| g0,2 |}+...+min{| gn−1,1 |+| gn−1,2 |}+min{| g0,1 0,2 n−1,1 |+| gn−1,2 |}] 2
where w = ( a|b) − ( a |b ) = ( a j − aj |b j − bj ) = (w j |wj ) (mod π), and d j (( a|b), ( a |b )) = wt j (w) where ( a|b) denotes 2n-tuple vector obtained by composed a and b.
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Let C be a code over Jπ2n . Then, the dual of C is defined as: C ⊥∗ = {( a|b) ∈ Jπ2n : ( a|b) ∗ ( a |b ) = 0
for all
( a |b ) ∈ C }
Here, ( a|b) ∗ ( a |b ) = Tr (ba − b a) where Tr : Jπr → Jπ is the trace map. Definition 3.1 We define unitary operators as X ( a)|u = |μ( a + u) and −1 Z (b) = |u = μ (bu) |u where a, b ∈ Jπ and is a primitive pth root of ∗ unity and μ : Fp → Jπ defined by μ( a) = a − [ aπp ]π. Hadamard gate is defined as H =
√1 ( as,t ) p
where as,t = (s−1)(t−1)modp for 1 ≤ s, t ≤ p.
Theorem 3.2 Let X ( a) Z (b) and X ( a ) Z (b ) be two operators such that X ( a)|u = −1 |μ( a + u) and Z (b) = |u = μ (bu) |u . The operators X ( a) Z (b) and X ( a ) Z (b ) are commutative if and only if μ−1 (b a) − μ−1 (ba ) = 0. −1
Proof. By Definition 3.1, we have μ (ab) X ( a) Z (b) = Z (b) X ( a). Let X ( a) Z (b) and ( a ) Z (b ) be two operators. The product of two operators is given by −1 X ( a) Z (b) X ( a ) Z (b ) = μ (ba ) X ( a) X ( a ) Z (b) Z (b ) and X ( a ) Z (b ) X ( a) Z (b) = −1 μ (b a) X ( a ) X ( a) Z (b ) Z (b). We already have X ( a) X ( a ) = X ( a ) X ( a) and Z (b) Z (b ) = Z (b ) Z (b). If X ( a) Z (b) and Z ( a ) Z (b ) commute, we obtain −1 −1 μ (ba ) = μ (b a) . This implies that μ−1 (b a) − μ−1 (ba ) = 0. Converse can be proved by using the similar argument. 2 The following theorem can be derived on page 450 of [4]. Theorem 3.3 Let C1 and C2 be two classical linear codes over Jπ with the parameters [n, k1 , d J1 ] and [n, k2 , d J2 ] such that C2 ⊆ C1 .Then,there exists an [[n, k1 − k2 , d J ]] quantum code with the minimum distance d J = min{d J1 , d J2 ⊥ }, where d J1 , d J2 ⊥ denotes the new minimum distance of the code C1 and C2 ⊥ over Jπ , respectively. ∗ aπ Example 3.4 Let p = 19, π = 5 + 2ρ and the map μ( a) = a − ππ ∗ π. We can find the image of each element of F19 under the map and obtain the following set: Jπ =5+2ρ = {0, 1, 2, −2 − 2ρ, −1 − 2ρ, −2ρ, −1 + ρ, ρ, 1 + ρ, 2 + ρ, −2 − ρ, −1 − ρ, −ρ, 1 − ρ, 2ρ, 1 + 2ρ, 2 + 2ρ, −2, −1}. Let (1 + ρ, 0, 1) ∈ Jπ3 be generator matrix of a code C. Then, C = {(0, 0, 0), (1 + ρ, 0, 1), (2 + 2ρ, 0, 2), (−2ρ, 0, −2 − 2ρ), (1 − ρ, 0, 1 − 2ρ), (2, 0, −2ρ), (−2 − ρ, 0, −1 + ρ), (−1, 0, ρ), (−1 − ρ, 0, 1 + ρ), (1 + 2ρ, 0, 2 + ρ), (−1 − 2ρ, 0, −2 − ρ), (−ρ, 0, −1 − ρ), (1, 0, −ρ), (2 + ρ, 0, 1 − ρ), (−2, 0, 2ρ), (−1 + ρ, 0, 1 + 2ρ), (2ρ, 0, 2 + 2ρ), (−2 − 2ρ, 0, −2), (−1 − ρ, 0, −1)}.
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Furthermore, the error basis can be obtained by the following formula X ( ak )i,j = δμ(i+k),μ( j) where ak ∈ Jπ . More detail can be found in [5].
4
Bounds for Quantum Codes over EJ-Integers
Parameters of a code is important in coding theory. One of the parameters is the minimum distance of the code and it is denoted by d. The distance d plays a significant role in error detection and correction abilities of a code. The more the value of d increases, the more the error detection and correction capability of the code also increase. However, finding such a d is always not easy. There are some lower and upper bounds for the codes. These bounds give us a idea about distance of the code. Bounds for quantum codes are little bit different from classical ones. For instance, quantum Hamming bound for an [[n, k, d]] code is given by d−2 1
∑
i =0
n i k 3 2 ≤ 2n . i
Here, the number of the information words is 2k and any error can occur at n different locations. In fact, in quantum case, errors can be represented by Pauli operators X, Y, Z and I and hence, there are 3i possible errors at every locations since identity operator I does not change any state. The total number of codewords which can be corrected cannot exceed the dimension of the code space, that is, 2n . Hence, we can obtain this inequality. If a code attains this bound, then it is called perfect code. Quantum singleton bound is also important since it contains an information about the distance of a code. For an [[n, k, d]] code, the bound becomes n − k + 2 ≥ 2d. Similar to classical codes, the codes which satisfying the quantum Singleton bound are called maximum distance separable quantum codes and they are known shortly as MDS codes. p −1
= −ρ
Let α1 , α2 be two elements of order p − 1 of Jπ such that α1 6 p −1 6
and α2 1)( x
p −1 6
and x x
p −1 2
= −ρ2 . Then, x
+ ρ)( x
p −1 6
p −1 6
p −1 2
+ 1 can be factored as x
+ ρ2 ) where x
p −1 6
p −1 2
+ 1 = (x
+ ρ = ( x + α1 )( x + α71 )...( x + ( p −6)
+ ρ2 = ( x + α2 )( x + α72 )...( x + α2
p −1 6
+
( p −6) α1 )
). If a polynomial divides
+ 1, then it is the generator polynomial of a code of length
p −1 2 .
Example 4.1 Let p = 31 and π = 5 + 6ρ. Let x2 − (4 + 4ρ) x + 7ρ and
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x3 − (5 + 3ρ) x2 + (8 + 11ρ) x − 7 − 14ρ be generator polynomials of C1 , C2 , respectively. Here, C2 ⊂ C1 . We can obtain a quantum code with parameters [[5, 1, 4]]π by using these two linear codes, since min{dC1 , dC⊥ } = 4. In 2 other respects, we can construct a nonbinary quantum code over F31 which is a field of characteristic 31. In a finite field, codes are constructed with respect to Hamming distance. When we consider quantum Singleton bound, the minimum distance of a code of length 5, size 1 cannot exceed 3, since the inequality n − k + 2 ≥ 2d holds. In fact, there exists a quantum [[5, 1, 3]]code and it attains the quantum Singleton bound. Hence, a quantum code which has the minimum distance greater than 3 cannot be obtained with respect to Hamming distance. This code with new parameters can correct some of errors of weight 2 and it takes an advantage for the correction procedure. Similarly, new codes over different fields can be obtained.
5
Conclusion
Existence of an isomorphism between a finite field of characteristic p and the set of EJ-integers with p elements provides us to construct quantum codes over EJ-integers. We defined a new distance for this new class of quantum codes, and we described error bases and matrices of them. Also, we proved the commutative property of error operators with respect to this new distance. As we see in Example 4.1 , obtaining these codes may lead an answer for the existence question for some new and better parameters.
References [1] Calderbank, R., and P. Shor,” Good Quantum Error-Correcting Codes Exist”, Physical Review 54 (1996), 1098-1105. [2] Dong, X., C. B. Soh, E. Gunawan and L. Tang, ”Groups of Algebraic Integers used for Coding QAM Signals”, Information Theory, IEEE 44 (1998), 1848 1860. [3] Huber, K., ”Codes over Eisenstein-Jacobi integers”, Contemporary Mathematics 168 (1994), 165-165 [4] Nielsen M. A., and I. L. Chuang, ”Quantum Computation and Quantum Information”, Cambridge University Press, Cambridge, 2002. [5] Yildiz, E., and F. Demirkale, ”Quantum codes over Eisenstein-Jacobi integers”, IEEEXplore, https://ieeexplore.ieee.org/document/7948934/.
Electronic Notes in Discrete Mathematics 70 (2018) 89–94
www.elsevier.com/locate/endm
On Bounds for Quantum Error Correcting Codes over EJ-Integers Eda YILDIZ 1 Department of Mathematics Yildiz Technical University Istanbul, Turkey
Abstract There are some differences between quantum and classical error corrections [4]. Hence, these differences should be considered when a new procedure is performed. In our recent study, we construct new quantum error correcting codes over different mathematical structures. The classical codes over Eisenstein-Jacobi(EJ) integers are mentioned in [3]. There is an efficient algorithm for the encoding and decoding procedures of these codes [3]. For coding over two-dimensional signal spaces like QAM signals, block codes over these integers p = 7, 13, 19, 31, 37, 43, 61, ... can be useful [2]. Thus, in this study, we introduce quantum error correcting codes over EJ-integers. This type of quantum codes may lead to codes with some new and good parameters. Keywords: Eisenstein-Jacobi integers, quantum codes, quantum error correcting codes, bounds on codes, QAM signals.
1
Email: [email protected]
https://doi.org/10.1016/j.endm.2018.11.015 1571-0653/© 2018 Elsevier B.V. All rights reserved.
90
1
E. Yildiz / Electronic Notes in Discrete Mathematics 70 (2018) 89–94
Introduction
Error correction is necessary during a data transmission. When a message is sent, some errors may occur in communication channel. Environmental effects, noises and the other interferences cause deformation on messages. Error correcting codes are used to correct these errors. While classical coding theory has been developing day by day, a new theory arised with an idea of a quantum computer. A new machine, which is based on quantum mechanics and called quantum computer, is expected to be more powerful than a classical computer. In 1976, Ingarden published a seminar paper and made a claim that classical information theory cannot be generalized directly to the quantum case. So, a new information theory based on quantum mechanics is needed. Hence, quantum error correction procedure is developed. The procedure is different and more complicated than the classical process. However, in 1996, a quantum code was invented by Robert Calderbank, Peter Shor and Andrew Steane. This code leads to a relation between classical linear codes and quantum codes. Thus, the problem of finding a quantum code transforms into finding a classical code containing the other code. This type of quantum code is known as CSS code. In general, there is no efficient decoding algorithm for block codes over two dimensional signal constellations such as QAM. So, different distances are used like Mannheim, Lee metric. It is better to develop decoding algorithm which works in an algebraic way for specific signal constellations. Huber, therefore, introduced linear codes over EJ-integers and defined a new two dimensional modular distance. He showed that there is an isomorphism between the set of EJ-integers with p element and finite field of characteristic p. The isomorphism leads to codes over these integers. Using of these codes may be more convenient for hexagonal signal constellations.
2
Preliminaries
In quantum channel, errors are represented by quantum operators. There are four important matrices X, Y, Z and I in quantum mechanics. They are called Pauli matrices and defined as follows: 01 0 −i 1 0 10 X = 1 0 , Y = i 0 , Z = 0 −1 , I = 0 1 In fact, any error can be represented as a linear combination of Pauli matrices. Since some features of these special matrices make it easier to calcu-
E. Yildiz / Electronic Notes in Discrete Mathematics 70 (2018) 89–94
91
late, they play an important role in the error correction procedure. Indeed, the set of Pauli matrices is not a group under the multiplication. However, it can be extended to a group, say P = {± X, ±iX, ±Y, ±iY, ± Z, ±iZ, ± I, ±iI }. Any Pauli operator on n qubits is a product of the elements of the group P. Moreover, a quantum code that encodes k qubits to n qubits is 2k -dimensional subspace of 2n -dimensional Hilbert space and it is denoted by [[n, k ]]. √
Definition 2.1 The set Z [ρ] = { a + bρ : a, b ∈ Z and ρ = −1+2 −3 } is a ring under the addition and multiplication. It is called ring of Eisentein-Jacobi integers and elements of the ring are called Eisenstein-Jacobi(EJ) integers. Primes of the form p ≡ 1 mod 6 can be written as sum of two integers such that p = a2 + 3b2 . Such primes are factored as ππ ∗ = p = a2 + 3b2 where π = a + b + 2b.ρ and π ∗ = a + b + 2b.ρ2 . There is a map μ : GF ( p) −→ Jπ defined by μ( a) = a −
aπ ∗ ππ ∗
π
where Jπ is the residue class of Z [ρ] modulo π and the operation [.] denotes rounding to the closest EJ integer. It is a one-to-one and onto homomorphism and hence, GF ( p) is isomorphic to Jπ . The isomorphism leads to a relation between a field of characteristic p and a set of EJ-integers with p elements. Here, each element of Jπ corresponds to a signal point. When each element of GF ( p) is represented as an element of Jπ , it provides important advantages for coding in the physical sense like hexagonal signal constellations. For the new codes, we also need to define a distance. The element α ∈ Jπ can be written in several different ways as α = g1 1 + g2 2 where 1 , 2 ∈ {±1, ±ρ, ±(1 + ρ)}. The weight of α is defined as w(α) = min{| g1 | + | g2 |} by Huber in [3]. The distance d between α1 and α2 is defined as d(α1 , α2 ) = w(α).
3
Quantum Codes over EJ Integers
Now, we will describe the structure of quantum codes over EJ-integers. Let p be a prime of the form p ≡ 1 mod 6. A p-ary quantum code C of length n and size k is a k-dimensional subspace of a pn -dimensional Hilbert space. We can define a new weight wt and a new distance d as follows: wt J (w) =
|+| g |}+...+ min {| g [min{| g0,1 |+| g0,2 |}+...+min{| gn−1,1 |+| gn−1,2 |}+min{| g0,1 0,2 n−1,1 |+| gn−1,2 |}] 2
where w = ( a|b) − ( a |b ) = ( a j − aj |b j − bj ) = (w j |wj ) (mod π), and d j (( a|b), ( a |b )) = wt j (w) where ( a|b) denotes 2n-tuple vector obtained by composed a and b.
E. Yildiz / Electronic Notes in Discrete Mathematics 70 (2018) 89–94
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Let C be a code over Jπ2n . Then, the dual of C is defined as: C ⊥∗ = {( a|b) ∈ Jπ2n : ( a|b) ∗ ( a |b ) = 0
for all
( a |b ) ∈ C }
Here, ( a|b) ∗ ( a |b ) = Tr (ba − b a) where Tr : Jπr → Jπ is the trace map. Definition 3.1 We define unitary operators as X ( a)|u = |μ( a + u) and −1 Z (b) = |u = μ (bu) |u where a, b ∈ Jπ and is a primitive pth root of ∗ unity and μ : Fp → Jπ defined by μ( a) = a − [ aπp ]π. Hadamard gate is defined as H =
√1 ( as,t ) p
where as,t = (s−1)(t−1)modp for 1 ≤ s, t ≤ p.
Theorem 3.2 Let X ( a) Z (b) and X ( a ) Z (b ) be two operators such that X ( a)|u = −1 |μ( a + u) and Z (b) = |u = μ (bu) |u . The operators X ( a) Z (b) and X ( a ) Z (b ) are commutative if and only if μ−1 (b a) − μ−1 (ba ) = 0. −1
Proof. By Definition 3.1, we have μ (ab) X ( a) Z (b) = Z (b) X ( a). Let X ( a) Z (b) and ( a ) Z (b ) be two operators. The product of two operators is given by −1 X ( a) Z (b) X ( a ) Z (b ) = μ (ba ) X ( a) X ( a ) Z (b) Z (b ) and X ( a ) Z (b ) X ( a) Z (b) = −1 μ (b a) X ( a ) X ( a) Z (b ) Z (b). We already have X ( a) X ( a ) = X ( a ) X ( a) and Z (b) Z (b ) = Z (b ) Z (b). If X ( a) Z (b) and Z ( a ) Z (b ) commute, we obtain −1 −1 μ (ba ) = μ (b a) . This implies that μ−1 (b a) − μ−1 (ba ) = 0. Converse can be proved by using the similar argument. 2 The following theorem can be derived on page 450 of [4]. Theorem 3.3 Let C1 and C2 be two classical linear codes over Jπ with the parameters [n, k1 , d J1 ] and [n, k2 , d J2 ] such that C2 ⊆ C1 .Then,there exists an [[n, k1 − k2 , d J ]] quantum code with the minimum distance d J = min{d J1 , d J2 ⊥ }, where d J1 , d J2 ⊥ denotes the new minimum distance of the code C1 and C2 ⊥ over Jπ , respectively. ∗ aπ Example 3.4 Let p = 19, π = 5 + 2ρ and the map μ( a) = a − ππ ∗ π. We can find the image of each element of F19 under the map and obtain the following set: Jπ =5+2ρ = {0, 1, 2, −2 − 2ρ, −1 − 2ρ, −2ρ, −1 + ρ, ρ, 1 + ρ, 2 + ρ, −2 − ρ, −1 − ρ, −ρ, 1 − ρ, 2ρ, 1 + 2ρ, 2 + 2ρ, −2, −1}. Let (1 + ρ, 0, 1) ∈ Jπ3 be generator matrix of a code C. Then, C = {(0, 0, 0), (1 + ρ, 0, 1), (2 + 2ρ, 0, 2), (−2ρ, 0, −2 − 2ρ), (1 − ρ, 0, 1 − 2ρ), (2, 0, −2ρ), (−2 − ρ, 0, −1 + ρ), (−1, 0, ρ), (−1 − ρ, 0, 1 + ρ), (1 + 2ρ, 0, 2 + ρ), (−1 − 2ρ, 0, −2 − ρ), (−ρ, 0, −1 − ρ), (1, 0, −ρ), (2 + ρ, 0, 1 − ρ), (−2, 0, 2ρ), (−1 + ρ, 0, 1 + 2ρ), (2ρ, 0, 2 + 2ρ), (−2 − 2ρ, 0, −2), (−1 − ρ, 0, −1)}.
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93
Furthermore, the error basis can be obtained by the following formula X ( ak )i,j = δμ(i+k),μ( j) where ak ∈ Jπ . More detail can be found in [5].
4
Bounds for Quantum Codes over EJ-Integers
Parameters of a code is important in coding theory. One of the parameters is the minimum distance of the code and it is denoted by d. The distance d plays a significant role in error detection and correction abilities of a code. The more the value of d increases, the more the error detection and correction capability of the code also increase. However, finding such a d is always not easy. There are some lower and upper bounds for the codes. These bounds give us a idea about distance of the code. Bounds for quantum codes are little bit different from classical ones. For instance, quantum Hamming bound for an [[n, k, d]] code is given by d−2 1
∑
i =0
n i k 3 2 ≤ 2n . i
Here, the number of the information words is 2k and any error can occur at n different locations. In fact, in quantum case, errors can be represented by Pauli operators X, Y, Z and I and hence, there are 3i possible errors at every locations since identity operator I does not change any state. The total number of codewords which can be corrected cannot exceed the dimension of the code space, that is, 2n . Hence, we can obtain this inequality. If a code attains this bound, then it is called perfect code. Quantum singleton bound is also important since it contains an information about the distance of a code. For an [[n, k, d]] code, the bound becomes n − k + 2 ≥ 2d. Similar to classical codes, the codes which satisfying the quantum Singleton bound are called maximum distance separable quantum codes and they are known shortly as MDS codes. p −1
= −ρ
Let α1 , α2 be two elements of order p − 1 of Jπ such that α1 6 p −1 6
and α2 1)( x
p −1 6
and x x
p −1 2
= −ρ2 . Then, x
+ ρ)( x
p −1 6
p −1 6
p −1 2
+ 1 can be factored as x
+ ρ2 ) where x
p −1 6
p −1 2
+ 1 = (x
+ ρ = ( x + α1 )( x + α71 )...( x + ( p −6)
+ ρ2 = ( x + α2 )( x + α72 )...( x + α2
p −1 6
+
( p −6) α1 )
). If a polynomial divides
+ 1, then it is the generator polynomial of a code of length
p −1 2 .
Example 4.1 Let p = 31 and π = 5 + 6ρ. Let x2 − (4 + 4ρ) x + 7ρ and
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94
x3 − (5 + 3ρ) x2 + (8 + 11ρ) x − 7 − 14ρ be generator polynomials of C1 , C2 , respectively. Here, C2 ⊂ C1 . We can obtain a quantum code with parameters [[5, 1, 4]]π by using these two linear codes, since min{dC1 , dC⊥ } = 4. In 2 other respects, we can construct a nonbinary quantum code over F31 which is a field of characteristic 31. In a finite field, codes are constructed with respect to Hamming distance. When we consider quantum Singleton bound, the minimum distance of a code of length 5, size 1 cannot exceed 3, since the inequality n − k + 2 ≥ 2d holds. In fact, there exists a quantum [[5, 1, 3]]code and it attains the quantum Singleton bound. Hence, a quantum code which has the minimum distance greater than 3 cannot be obtained with respect to Hamming distance. This code with new parameters can correct some of errors of weight 2 and it takes an advantage for the correction procedure. Similarly, new codes over different fields can be obtained.
5
Conclusion
Existence of an isomorphism between a finite field of characteristic p and the set of EJ-integers with p elements provides us to construct quantum codes over EJ-integers. We defined a new distance for this new class of quantum codes, and we described error bases and matrices of them. Also, we proved the commutative property of error operators with respect to this new distance. As we see in Example 4.1 , obtaining these codes may lead an answer for the existence question for some new and better parameters.
References [1] Calderbank, R., and P. Shor,” Good Quantum Error-Correcting Codes Exist”, Physical Review 54 (1996), 1098-1105. [2] Dong, X., C. B. Soh, E. Gunawan and L. Tang, ”Groups of Algebraic Integers used for Coding QAM Signals”, Information Theory, IEEE 44 (1998), 1848 1860. [3] Huber, K., ”Codes over Eisenstein-Jacobi integers”, Contemporary Mathematics 168 (1994), 165-165 [4] Nielsen M. A., and I. L. Chuang, ”Quantum Computation and Quantum Information”, Cambridge University Press, Cambridge, 2002. [5] Yildiz, E., and F. Demirkale, ”Quantum codes over Eisenstein-Jacobi integers”, IEEEXplore, https://ieeexplore.ieee.org/document/7948934/.