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1 Journal of Physics: Conference Series PAPER OPEN ACCESS Quantum codes from cyclic codes over F 3 + μf 3 + υf 3 + μυ F 3 To cite this article: Mehmet Özen et al 2016 J. Phys.: Conf. Ser Recent citations - u-constacyclic codes over $${\mathbb {F}}_p+u{\mathbb {F}}_p$$ F p + u F p and their applications of constructing new non-binary quantum codes Jian Gao and Yongkang Wang View the article online for updates and enhancements. This content was downloaded from IP address on 03/09/2018 at 15:26

2 Quantum codes from cyclic codes over F 3 + uf 3 + vf 3 + uvf 3 Mehmet Özen 1, N. Tuğba Özzaim 1 and Halit İnce 1 1 Department of Mathematics, Sakarya University, Sakarya, Turkey ozen@sakarya.edu.tr, tugbaozzaim@gmail.com, ince@sakarya.edu.tr Abstract. In this paper, it will be argued that structure of cyclic codes over F 3 + uf 3 + vf 3 + uvf 3 where u 2 = 1, v 2 = 1 and uv = vu for arbitrary length n. We define a new Gray map which is a distance preserving map. By using decomposition theory, we find generator polynomials of cyclic codes over F 3 + uf 3 + vf 3 + uvf 3 and obtain the result that cyclic codes over F 3 + uf 3 + vf 3 + uvf 3 are principally generated. Further, using these results we determine the parameters of quantum codes which constructed from cyclic codes over F 3+uF 3+vF 3+uvF 3. We present some of the results of computer search. 1. Introduction Quantum error-correction plays an important role in quantum computing since it is used to correct any errors in quantum data due to decoherence and other quantum noise. Firstly, the existence of quantum error correcting code was shown by Shor [1] and independently by Steane [2]. In 1998, Calderbank et. al. published a paper in which they developed the theory to construct quantum codes by using classical error correcting codes. In recent years, a considerable literature has grown up around the quantum error correcting codes. Some authors constructed quantum codes by using the Gray image of cyclic codes over some finite rings. For example, in [7], a new method constructing quantum codes from cyclic codes over finite ring F 2 + vf 2 where v 2 = v is given by Qian. In [4], Kai and Zhu construct quantum codes from Hermitian self orthogonal codes over F 4 as Gray images of linear and cyclic codes over F 4 + uf 4 where u 2 = 0. In [5], Yin and Ma give an existence condition of quantum codes which are obtained from cyclic codes over F 2 + uf 2 + u 2 F 2 with Lee metric. In [6], some nonbinary quantum codes using classical codes over Gaussian integers are obtained by Ozen et. al. In [9], Guenda et. al. extend the CSS construction to finite commutative Frobenius rings. Dertli et. al. obtain quantum codes from cyclic codes over F 2 + uf 2 + vf 2 + uvf 2 in [10]. In [8], Ashraf and Mohammad give a construction for quantum codes from cyclic codes over F 3 + vf 3 where v 2 = 1. This ring is motivation to construct a new ring F 3 + uf 3 + vf 3 + uvf 3 where u 2 = 1 and v 2 = 1 because an advantage of the ring F 3 + uf 3 + vf 3 + uvf 3 is that it allows to find more parameters of quantum codes over F 3. This paper has been organised in the following way. The second section is concerned with the preliminaries used for this study. The third section presents a new Gray map from F 3 +uf 3 +vf 3 +uvf 3 to F 4 3 and linear codes over F 3 +uf 3 +vf 3 +uvf 3 and their properties. In Section 4, we investigate the structure of cyclic codes over F 3 +uf 3 +vf 3 +uvf 3. In Section 5, we give condition for cyclic codes over F 3 +uf 3 +vf 3 +uvf 3 to contain their dual and determine the Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by Ltd 1

3 parameters of quantum codes which constructed from cyclic codes over F 3 + uf 3 + vf 3 + uvf 3. At the end of the paper, we give an example and a table to illustrate the results. 2. Preliminaries Let F 3 = {0, 1, 2} be a finite field and R denotes the ring F 3 +uf 3 +vf 3 +uvf 3 = {a+ub+vc+uvd : a, b, c, d F 3 } where u 2 = 1, v 2 = 1 and uv = vu. This ring is a principal ideal ring but not a chain ring and the ideals < 2 + u + 2v >, < 2 + v + uv >, < u + v + 2uv >, < 2u + 2v + 2uv > are maximal ideals of F 3 + uf 3 + vf 3 + uvf 3. Let C be a nonempty subset of R n. If C is a R-submodule of R n, then C is called a linear code of length n over R. For any a = (a 0, a 1,..., a n 1 ) and b = (b 0, b 1,..., b n 1 ) F 3 the inner product is given by a.b = a 0 b 0 + a 1 b a n 1 b n 1. If a.b = 0 then a and b are said to be orthogonal. Let C be a linear code of length n over R. Then the dual code of C is C = {a R a.b = 0, b C}. C is said to be self orthogonal if C C and self dual if C = C. A linear code C of length n is called cyclic if τ(c) = (c n 1, c 0,..., c n 2 ) C,where τ is cyclic shift operator, whenever c = (c 0, c 1,..., c n 1 ) C. Let R n denote the quotient ring R[x]/ x n 1. Then we consider the following correspondence: φ : R n c = (c 0, c 1,..., c n 1 ) R n c 0 + c 1 x c n 1 x n 1 = c(x). It is well known that C is a cyclic code if and only if φ(c) is an ideal of R n. Let c = (c 0, c 1,..., c n 1 ) be a codeword in C. The Hamming weight of c = (c 0, c 1,..., c n 1 ) is the number of nonzero coordinates in c and is denoted by w H (c). The Hamming distance between two elements c 1, c 2 C is defined as d H (c 1, c 2 ) = ω H (c 1 c 2 ). Let α (F n 3 )l with α = (α 1, α 2,..., α nl ) = (α (1) α (2)... α (l) ). Let ϕ l be the quasi cyclic shift on (F n 3 )l given by ϕ l (α (1) α (2)... α (l) ) = (τ(α (1) ) τ(α (2) )... τ(α (l) )). A code of length nl over F 3 is called as quasi cyclic code of index l if ϕ l (C) = C. 3. Gray map and linear codes over F 3 + uf 3 + vf 3 + uvf Gray map Any element of R can be written as a + vb + uc + uvd R, where a, b, c, d F 3. The Gray map from R to F 4 3 is defined as follows: ψ : R F 4 3 a + vb + uc + uvd (a + b + c + d, a b + c d, a + b c d, a b c + d). This map can be naturally extended from R n to F 4n 3. We define the Lee weight of any x R as ω L (x) = ω H (ψ(x)). For any elements x, y R, the Lee distance is given by d L (x, y) = ω L (x y). Theorem 3.1. The Gray map is an isometry from (R n, Lee distance) to (F 4n 3, Hamming distance). Proof. For any x 1, x 2 R and λ F 3, ψ(x 1 + x 2 ) = ψ(x 1 ) + ψ(x 2 ) and ψ(λx 1 ) = λψ(x 1 ). So ψ is linear. Now we show that ψ is distance preserving map. From the definition, we have d L (x 1, x 2 ) = ω L (x 1 x 2 ) = ω H (ψ(x 1 x 2 )) = ω H (ψ(x 1 ) ψ(x 2 )) = d H (ψ(x 1 ) ψ(x 2 )). Theorem 3.2. If C is a linear codes of length n over F 3 + uf 3 + vf 3 + uvf 3 with C = 3 k and minimum Lee weight d L then ψ(c) is [4n, k, d H ] ternary linear codes over F 3. Theorem 3.3. If C is self orthogonal then ψ(c) is self orthogonal. 2

4 Proof. Let C be self orthogonal and α = a + vb + uc + uvd, β = x + vy + uz + uvt C where a, b, c, d, x, y, z, t F 3. Since C is self orthogonal, we have α.β = ax + by + cz + dt + v(ay + bx + ct + dz) + u(az + bt + cx + dy) + uv(at + bz + cy + dx) = 0. So we have ax + by + cz + dt = (ay + bx + ct + dz) = (az + bt + cx + dy) = (at + bz + cy + dx) = 0. Then the other side ψ(α).ψ(β) = (a + b + c + d, a b + c d, a + b c d, a b c + d)(x + y + z + t, x y + z t, x + y z t, x y z + t) = 0. Hence ψ(c) is self orthogonal Linear codes over F 3 + uf 3 + vf 3 + uvf 3 Let A 1, A 2, A 3, A 4 are linear codes, then we denote that A 1 A 2 A 3 A 4 = {a 1 +a 2 +a 3 +a 4 : a i A i ; 1 i 4} and A 1 A 2 A 3 A 4 = {(a 1, a 2, a 3, a 4 ) : a i A i ; 1 i 4}. Definition 3.4. Let C be a linear code of length n over R. Define C 1 = {x + y + z + t F 3 x + vy + uz + uvt C} C 2 = {x y + z t F 3 x + vy + uz + uvt C} C 3 = {x + y z t F 3 x + vy + uz + uvt C} C 4 = {x y z + t F 3 x + vy + uz + uvt C} Theorem 3.5. Let C be a linear code of length n over R. Then ψ(c) = C 1 C 2 C 3 C 4 and C = C 1. C 2. C 3. C 4. Proof. For any (x 1,..., x n, y 1,..., y n, z 1,..., z n, t 1,..., t n ) ψ(c), let c i = x i + y i + z i + t i + v(x i y i + z i t i ) + u(x i + y i z i t i ) + uv(x i y i z i + t i ) ; 1 i n. Since ψ is bijection c = (c 1, c 2,..., c n ) C. By the definition of C 1, C 2, C 3, C 4 we have (x 1, x 2..., x n ) C 1, (y 1, y 2,..., y n ) C 2, (z 1, z 2,..., z n ) C 3, (t 1, t 2,..., t n ) C 4. So, (x 1,..., x n, y 1,..., y n, z 1,..., z n, t 1,..., t n ) C 1 C 2 C 3 C 4. It means that ψ(c) C 1 C 2 C 3 C 4. On the other hand, let (x 1,..., x n, y 1,..., y n, z 1,..., z n, t 1,..., t n ) C 1 C 2 C 3 C 4 where x = (x 1, x 2..., x n ) C 1, y = (y 1, y 2,..., y n ) C 2, z = (z 1, z 2,..., z n ) C 3, t = (t 1, t 2,..., t n ) C 4. There are a = (a 1, a 2,..., a n ), b = (b 1, b 2,..., b n ), c = (c 1, c 2,..., c n ), d = (d 1, d 2,..., d n ) C such that a i = x i + (1 + 2v + u + 2uv)p i, b i = y i + (1 + v + u + uv)q i, c i = z i + (1 + 2v + 2u + uv)r i, d i = t i + (1 + v + 2u + 2uv)s i where p i, q i, r i, s i F 3. Since C is linear then c = (1 + v + u + uv)a + (1 + 2v + u + 2uv)b + (1 + v + 2u + 2uv)c + (1 + 2v + 2u + uv)d = x + y + z + t + (x y + z t)v + (x + y z t)u + (x y z + t)uv. It follows that ψ(c) = (x, y, z, t) ψ(c). So C 1 C 2 C 3 C 4 ψ(c). Therefore ψ(c) = C 1 C 2 C 3 C 4. Moreover since ψ(c) is bijection, C = ψ(c) = C 1 C 2 C 3 C 4 = C 1. C 2. C 3. C 4. The rest of the paper, the terms λ 1, λ 2, λ 3, λ 4 will refer to (1+v +u+uv), (1+2v +u+2uv), (1 + v + 2u + 2uv), (1 + 2v + 2u + uv) respectively. Lemma 3.6. For i = 1, 2, 3, 4, if G i is generator matrix of ternary linear codes C i, then the λ 1 G 1 generator matrix of C is λ 2 G 2 λ 3 G 3. λ 4 G 4 Proof. If G i is generator matrix of ternary linear code C i then the generator matrix of G ψ(c) = C 1 C 2 C 3 C 4 is 0 G G 3 0. By the Theorem 3.5, the generator G 4 matrix of C is λ 1 G 1 λ 2 G 2 λ 3 G 3 λ 4 G 4. 3

5 Corollary 3.7. If ψ(c) = C 1 C 2 C 3 C 4, then C = λ 1 C 1 λ 2 C 2 λ 3 C 3 λ 4 C 4. Lemma 3.8. Let d H and d L denote the Hamming and Lee distance respectievly. Then d H = d L = min{d(c 1 ), d(c 2 ), d(c 3 ), d(c 4 )}. Proof. Since ψ distance preserving map we obtain d L (C) = d H (ψ(c)) = d H (C 1 C 2 C 3 C 4 ) = min{d(c 1 ), d(c 2 ), d(c 3 ), d(c 4 )}. 4. Cyclic codes over F 3 + uf 3 + vf 3 + uvf 3 Theorem 4.1. Let C = λ 1 C 1 λ 2 C 2 λ 3 C 3 λ 4 C 4 be a linear code of length n over R. Then C is a cyclic code of length n over R if and only if for i = 1, 2, 3, 4, C i is a cyclic code of length n over F 3. Proof. Let c = (c 1, c 2,..., c n ) C where c j = λ 1 x j + λ 2 y j + λ 3 z j + λ 4 t i j = 1,..., n and x = (x 1, x 2..., x n ) C 1, y = (y 1, y 2,..., y n ) C 2, z = (z 1, z 2,..., z n ) C 3, t = (t 1, t 2,..., t n ) C 4. Since for i = 1, 2, 3, 4, C i is a cyclic code of length n over F 3 we get τ(x) = (x n, x 1,..., x n 1 ) C 1, τ(y) = (y n, y 1,..., y n 1 ) C 2, τ(z) = (z n, z 1,..., z n 1 ) C 3, τ(t) = (t n, t 1,..., t n 1 ) C 4. Note that τ(c) = (c n, c 1,..., c n 1 ) = λ 1 τ(x) + λ 2 τ(y) + λ 3 τ(z) + λ 4 τ(t) C which means that C is a cyclic code over R. On the other side, let x = (x 1, x 2..., x n ) C 1, y = (y 1, y 2,..., y n ) C 2, z = (z 1, z 2,..., z n ) C 3, t = (t 1, t 2,..., t n ) C 4. Assume that c = (c 1, c 2,..., c n ) C where c j = λ 1 x j +λ 2 y j +λ 3 z j + λ 4 t i for j = 1,..., n. Since C is a cyclic code over R then we have τ(c) = (c n, c 1,..., c n 1 ) C. Note that again (c n, c 1,..., c n 1 ) = λ 1 (x n,..., x n 1 ) + λ 2 (y n,..., y n 1 ) + λ 3 (z n,..., z n 1 ) + λ 4 (t n,..., t n 1 ) C. Hence τ(x) C 1 τ(y) C 2 τ(z) C 3 τ(t) C 4. That means for i = 1, 2, 3, 4, C i is a cyclic code of length n over F 3. Now we investigate generator polynomial of cyclic codes of length n over R. For i = 1, 2, 3, 4 since C i is a cyclic code over F 3, we make use of the generators of cyclic codes over F 3. Theorem 4.2. Let C = λ 1 C 1 λ 2 C 2 λ 3 C 3 λ 4 C 4 be a cyclic code of length n over R. Then there exist the polynomials g i (x) s,1 i 4, over F 3 [x] such that C = < λ 1 g 1 (x), λ 2 g 2 (x), λ 3 g 3 (x), λ 4 g 4 (x) > where g i (x) s are generator polynomials of C i 4 s for i = 1, 2, 3, 4. Moreover C = 3 4n i=1 deg(g i (x)). Proof. Let C i = < g i (x) > and C = λ 1 C 1 λ 2 C 2 λ 3 C 3 λ 4 C 4 then C = {c(x) = λ 1 g 1 s 1 + λ 2 g 2 s 2 + λ 3 g 3 s 3 + λ 4 g 4 s 4 s i F 3 [x]}. It is easily seen that C < λ 1 g 1 (x), λ 2 g 2 (x), λ 3 g 3 (x), λ 4 g 4 (x) >. On the other hand, let d(x) < λ 1 g 1 (x), λ 2 g 2 (x), λ 3 g 3 (x), λ 4 g 4 (x) > then for 1 i 4, there is k i (x) R[x] such that d(x) = λ 1 g 1 (x)k 1 (x) + λ 2 g 2 (x)k 2 (x) + λ 3 g 3 (x)k 3 (x) + λ 4 g 4 (x)k 4 (x). Then there exists polynomial s i (x) F 3 [x] such that λ 1 s 1 (x) = λ 1 k 1 (x), λ 2 s 2 (x) = λ 2 k 2 (x), λ 3 s 3 (x) = λ 3 k 3 (x), λ 4 s 4 (x) = λ 4 k 4 (x). Hence we have < λ 1 g 1 (x), λ 2 g 2 (x), λ 3 g 3 (x), λ 4 g 4 (x) > C. This gives that C = < λ 1 g 1 (x), λ 2 g 2 (x), λ 3 g 3 (x), λ 4 g 4 (x) >. From Theorem 3.5 we know that C = C 1. C 2. C 3. C 4. Thus we get C = 3 4 4n deg(g i (x)) i=1. 4

6 Theorem 4.3. Let C be a cyclic code of length n over R, then there is a polynomial g(x) such that C = < g(x) > where g(x) = λ 1 g 1 (x) + λ 2 g 2 (x) + λ 3 g 3 (x) + λ 4 g 4 (x) and g(x) x n 1. Proof. By Theorem 4.2, we know that C = < λ 1 g 1 (x), λ 2 g 2 (x), λ 3 g 3 (x), λ 4 g 4 (x) > where g i (x) is generator polynomial of C i for i = 1, 2, 3, 4. Let g(x) = λ 1 g 1 (x) + λ 2 g 2 (x) + λ 3 g 3 (x) + λ 4 g 4 (x), obviously < g(x) > C. On the other side, note that λ 1 g(x) = λ 1 g 1 (x), λ 2 g(x) = λ 2 g 2 (x), λ 3 g(x) = λ 3 g 3 (x), λ 1 g(x) = λ 1 g 1 (x) then this gives C < g(x) >. Hence C = < g(x) >. Since g i (x) x n 1 there exists r i (x) R[x]/x n 1 such that x n 1 = g 1 (x)r 1 (x) = g 2 (x)r 2 (x) = g 3 (x)r 3 (x) = g 4 (x)r 4 (x). It follows that x n 1 = g(x)[λ 1 r 1 (x)+λ 2 r 2 (x)λ 3 g 3 (x)r 3 (x)+λ 4 g 4 (x)r 4 (x)], hence g(x) x n 1. Corollary 4.4. Every ideal of R[x]/x n 1 is principal. Theorem 4.5. Let ψ be the gray map and ϕ 4 and τ be quasi and cyclic shift operator as in the preliminaries. Then ψτ = ϕ 4 ψ. Proof. The proof is directly from the definition of quasi and cyclic codes. Theorem 4.6. C is a cyclic code of length n over R if and only if ψ(c) is a quasi cyclic code of index 4 over F 3 with length 4n. Proof. Let C be a cyclic code of length n over R i.e. τ(c) = C. Using Theorem 4.5 we have ϕ 4 (ψ(c)) = ϕ(τ(c)) = ψ(c). Hence ψ(c) is a quasi cyclic code of index 4 over F 3. On the other hand, let ψ(c) is a quasi cyclic codes of index 4 over F 3 i.e. ϕ 4 (ψ(c)) = ψ(c) then using Theorem 4.5 again we get ψ(τ(c)) = ϕ 4 (ψ(c)) = ψ(c). Note that ψ is injective so τ(c) = C. 5. Quantum codes from cyclic codes over F 3 + uf 3 + vf 3 + uvf 3 Theorem 5.1. Let C and Ĉ be two linear codes with parameters [n, k, d] and [n, ˆk, ˆd] such that C Ĉ then there exist a quantum code with parametre [n, k + ˆk, min{d, ˆd}]. Especially if C C then there exists an [[n,2k-n,d]] quantum code. Theorem 5.2. Let C = λ 1 C 1 λ 2 C 2 λ 3 C 3 λ 4 C 4 be a cyclic code of length n over R. Then C =< λ 1 h 1(x) + λ 2 h 2(x) + λ 3 h 3(x) + λ 4 h 4(x) > and C = 3 deg g 1(x)+deg g 2 (x)+deg g 3 (x)+deg g 4 (x) where for i = 1, 2, 3, 4, h i (x) are reciprocal polynomials of h i (x) i.e. h i (x) = x n 1/g i (x), h i (x) = xdeg hi(x) h i (x 1 ). Lemma 5.3. A cyclic code C with generator polynomial g(x) contains its dual if and only if where g (x) is reciprocal polynomial of g(x). x n 1 0 (mod g(x)g (x)) Theorem 5.4. C = < λ 1 g 1 (x), λ 2 g 2 (x), λ 3 g 3 (x), λ 4 g 4 (x) > be a cyclic code of length n over R. Then C C if and only if x n 1 0 (mod g i gi ) for i = 1, 2, 3, 4. Proof. For i = 1, 2, 3, 4, let x n 1 0 (mod g i gi ) then from Lemma 5.3, we have C i C i. This implies that λ i Ci λ i C i. Hence < λ 1 h 1(x) + λ 2 h 2(x) + λ 3 h 3(x) + λ 4 h 4(x) > < λ 1 g 1 (x), λ 2 g 2 (x), λ 3 g 3 (x), λ 4 g 4 (x) >. Thus C C. Conversely, if C C then λ 1 C 1 λ 2 C 2 λ 3 C 3 λ 4 C 4 λ 1 C 1 λ 2 C 2 λ 3 C 3 λ 4 C 4. Since C i is a ternary code such that λ i C i is equal to C mod λ i. Thus Ci C i and x n 1 0 (mod g i gi ) for i = 1, 2, 3, 4. 5

7 Corollary 5.5. Let C = λ 1 C 1 λ 2 C 2 (λ 3 C 3 λ 4 C 4 be a cyclic code of length n over R. Then C C if and only if C i C i for i = 1, 2, 3, 4. By Theorem 5.1 and Corollary 5.5, the quantum codes can be constructed. Theorem 5.6. Let C = λ 1 C 1 λ 2 C 2 λ 3 C 3 λ 4 C 4 be a cyclic code of length n over R with size 3 k. If C C and there exist a quantum code with parameters [[4n, 2k 4n, d L ]]. Example 5.7. Let n = 18. Then x 18 1 = (x + 1) 9 (x + 2) 9 in F 3. Let g(x) = λ 1 g 1 (x) + λ 2 g 2 (x) + λ 3 g 3 (x) + λ 4 g 4 (x) where g 1 (x) = g 2 (x) = g 3 (x) = x 5 + 2x 3 + 2x and g 4 (x) = x 6 + x 5 + 2x 4 + x 3 + 2x 2 + x + 1 so that g1 (x) = g 2 (x) = g 3 (x) = x5 + 2x 3 + 2x g4 (x) = x6 + x 5 + 2x 4 + x 3 + 2x 2 + x + 1. Let C = < g(x) > be a cyclic code over R. Clearly x 18 1 is divisible by g i gi for i = 1, 2, 3, 4. Hence by Theorem 5.5 we have C C. Then a quantum code with parameters [[72, 30, 3]] can be obtained. Table 1. The parameters of quantum code. n Generator polynomials [[N, K, D]] 6 g 1 = g 2 = x + 1 and g 3 = g 4 = x + 2 [[24, 16, 2]] 9 g 1 = g 2 = g 3 = g 4 = x + 2 [[36, 28, 2]] 11 g 1 = g 2 = x 5 + x 4 + 2x 3 + x and g 3 = g 4 = x 5 + 2x 3 + x 2 + 2x + 2 [[44, 4, 5]] 15 g 1 = g 2 = g 3 = g 4 = x + 2 [[60, 52, 2]] 18 g 1 = g 2 = g 3 = g 4 = x + 2 [[72, 64, 2]] 21 g 1 = g 2 = g 3 = x + 2 and g 4 = x + 1 [[84, 76, 2]] 22 g 1 = g 2 = g 3 = g 4 = x 10 + x 9 + x 8 + 2x 7 + 2x 6 + 2x 5 + x 4 + 2x 3 + 2x 2 + x + 2 [[88, 8, 7]] 23 g 1 = g 2 = g 3 = g 4 = x 11 + x 10 + x 9 + 2x 8 + 2x 7 + x 5 + x [[92, 4, 8]] 24 g 1 = g 2 = g 3 = g 4 = x 5 + x 3 + 2x + 2 [[96, 56, 3]] 30 g 1 = g 2 = g 3 = x + 2 and g 4 = x [[120, 110, 2]] 35 g 1 = g 2 = g 3 = g 4 = x x x 10 + x 9 + 2x 8 + x 7 + x 5 + 2x 4 + x [[140, 44, 5]] References [1] P. W. Shor, Scheme for reducing decoherence in quantum computer memory, Phys. Rev. A, 52 (1995), [2] A. M. Steane, Simple quantum error correcting codes, Phys. Rev. Lett., 77 (1996), [3] A. R. Calderbank, E.M.Rains, P.M.Shor, N.J.A.Sloane, Quantum error correction via codes over GF (4), IEEE Trans. Inf. Theory, 44 (1998), [4] X.Kai,S.Zhu, Quaternary construction of quantum codes from cyclic codes over F 4 + vf 4, Int. J. Quantum Inform., 9 (2011), [5] X. Yin, W. Ma, Gray Map And Quantum Codes Over The Ring F 2 + uf 2 + u 2 F 2; International Joint Conferences of IEEE TrustCom-11, (2011). [6] M. Ozen, M. Guzeltepe, Quantum Codes From Codes over Gaussian Integers with Respect to The Mannheim Metric,Quantum Information and Computation, 12 (2012), [7] J.Qian, Quantum codes from cyclic codes over F 2 + vf 2, Journal of Inform. Computational Science 10 : 6 (2013), [8] M. Ashraf, G. Mohammad, Quantum codes from cyclic codes over F 3 +vf 3, International Journal of Quantum Information, vol. 12, No. 6(2014) [9] K. Guenda, T. A. Gulliver, Quantum codes over rings, Int. J. Quantum Inform, 12, (2014), [10] A. Dertli, Y. Cengellenmis, S. Eren, On quantum codes obtained from cyclic codes over A 2, Int. J. Quantum Inform., vol. 13, 2(2015)

ON QUANTUM CODES FROM CYCLIC CODES OVER A CLASS OF NONCHAIN RINGS

Bull Korean Math Soc 53 (2016), No 6, pp 1617 1628 http://dxdoiorg/104134/bkmsb150544 pissn: 1015-8634 / eissn: 2234-3016 ON QUANTUM CODES FROM CYCLIC CODES OVER A CLASS OF NONCHAIN RINGS Mustafa Sari