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For any odd prime $p$ , and a divisor $\ell $ of $p$ , we consider $I_\ell $ to be the set of all divisors of $p-1$ , which are less than or equal to $\ell $. In this letter, we construct quantum codes from cyclic codes over $\mathbb F_{p}$ and $\mathbb F_{p} S_\ell $ , where $S_\ell =\prod _{i\in I_\ell }R_{i}$ , for $R_{i}=\frac {\mathbb F_{p}[u]}{\langle u^{i+1}-u\rangle }$. For that, first we construct linear codes and a Gray map over $R_\ell $. Using this construction, we study cyclic codes over $R_\ell $ , and then extend that over $\mathbb F_{p} S_\ell $. We also give a Gray map over $\mathbb F_{p} S_\ell $. Then, using necessary and sufficient condition of dual containing property for cyclic codes, we construct quantum MDS codes. It is observed that the quantum codes constructed are new in the literature. [ABSTRACT FROM AUTHOR]

In this paper we introduce a new class of sparse CSS quantum error-correcting codes based on new orientable self-dual embeddings of graphs. Each code in this class is associated with the embedding of the surface so that the qubits correspond to the edges of the embedding. The parameters of the new codes are [ [ m (m + 1) 2 , m (m - 3) 2 , 3 ] ] , where m = 4 s , s ≥ 1 . We also present a table of quantum codes whose parameters had not been shown before. [ABSTRACT FROM AUTHOR]

Let p be an odd prime, s and m be positive integers. Cyclic codes of length $2p^{{s}}$ over $\mathbb {F}_{{p}^{{m}}}$ are the ideals $\langle ({x}-1)^{i}({x}+1)^{j} \rangle $ , where $0 \le {i}, {j} \le {p}^{ {s}}$ , of the principal ideal ring $\mathbb {F}_{{p}^{{m}}}[{x}]/\langle {x}^{2\textit {p}^{\textit {s}}}-1 \rangle $. Using this structure, the symbol-pair distances of all cyclic codes of length $2\textit {p}^{\textit {s}}$ over $\mathbb F_{{p}^{{m}}}$ are completely determined. In addition, we establish all MDS symbol-pair cyclic codes of length $2\textit {p}^{\textit {s}}$ over $\mathbb F_{{p}^{{m}}}$. Some MDS symbol-pair cyclic codes are better than all the known ones. Among others, we discuss possible applications to construct quantum MDS symbol-pair codes. [ABSTRACT FROM AUTHOR]

In this paper, we consider a family of one-generator quasi-cyclic codes and their applications in quantum codes construction. We give a sufficient condition for self-orthogonality with respect to Hermitian inner product. By virtue of the well-known MacWilliams equations, some binary and ternary stabilizer quantum codes with good parameters are constructed. Furthermore, we present a lower bound on the Hermitian dual distance of these involved codes. As the computational results, some good stabilizer quantum codes over small finite fields are obtained. [ABSTRACT FROM AUTHOR]

It is an important task to construct quantum maximum-distance-separable (MDS) codes with good parameters. In the present paper, we provide six new classes of $q$ -ary quantum MDS codes by using generalized Reed–Solomon (GRS) codes and Hermitian construction. The minimum distances of our quantum MDS codes can be larger than $\frac {q}{2}+1$. Three of these six classes of quantum MDS codes have longer lengths than the ones constructed in and , hence some of their results can be easily derived from ours via the propagation rule. Moreover, some known quantum MDS codes of specific lengths can be seen as special cases of ours and the minimum distances of some known quantum MDS codes are also improved as well. [ABSTRACT FROM AUTHOR]

Let $q$ be a prime power and $m\geq 3$ be odd. Suppose that $n=\frac {q^{2m}-1}{a}$ with $a|(q^{m}+1)$ and $3\leq a \leq 2(q^{2}-q+1)$. This paper mainly determines the actual maximum designed distance of Hermitian dual-containing Bose-Chaudhuri-Hocquenghem (BCH) codes over $\mathbb {F}_{q^{2}}$ of length $n$. Firstly, we give the maximum designed distance $\delta _{m,a}^{R}$ of narrow-sense Hermitian dual-containing BCH codes. Secondly, we show that there are also non-narrow-sense ones of designed distance up to $\delta _{m,a}^{R}$. It is worth mentioning that our maximum designed distance $\delta _{m,a}^{R}>\lceil \frac {a}{2}\rceil \delta _{m}^{A}$ , where $\delta _{m}^{A}$ is given by Aly et al. (IEEE Trans. Inf. Theory, vol. 53, no. 3, pp. 1183-1188, 2007). Thus, many families of Hermitian dual-containing BCH codes with relatively large designed distance are obtained. Using the Hermitian construction to them, we can subsequently construct different classes of nonprimitive quantum codes, which are new in the sense that their parameters are not covered in the literature. [ABSTRACT FROM AUTHOR]

In this paper we construct two classes of binary quantum error-correcting codes on closed orientable surfaces. These codes are derived from self-dual orientable embeddings of complete bipartite graphs and complete multipartite graphs on the corresponding closed orientable surfaces. We also show a table comparing the rate of these quantum codes when fixing the minimum distance to 3 and 4. [ABSTRACT FROM AUTHOR]

In this paper, we introduce self-dual cyclic codes over the ring ℤ 2 × (ℤ 2 + u ℤ 2) = ℤ 2 ℤ 2 [ u ]. We determine the conditions for any ℤ 2 ℤ 2 [ u ] -cyclic code to be self-dual, that is, 𝒞 = 𝒞 ⊥ . Since the binary image of a self-orthogonal ℤ 2 ℤ 2 [ u ] -linear code is also a self-orthogonal binary linear code, we introduce quantum codes over ℤ 2 ℤ 2 [ u ]. Finally, we present some examples of self-dual cyclic and quantum codes that have good parameters. [ABSTRACT FROM AUTHOR]

Commonly classical error correcting codes received from a hyperelliptic curves that have great parameters. Although, the quantum stabilizer codes received from this classical error correcting codes via Euclidean or Hermitian selforthogonality do not continuously have great parameters. In this paper, the Hermitian selforthogonality of algebraic geometry codes received from one hyperelliptic curves was explored. it is obvious that the quantum codes produced from such Hermitian selforthogonal classical codes have great parameters. [ABSTRACT FROM AUTHOR]

In this work, skew quasi cyclic codes over R = 𝔽 q + v 𝔽 q , where v 2 = v are considered. The generating set for one generator skew quasi cyclic codes over R is also determined. We discuss a sufficient condition for one generator skew quasi cyclic codes to be free. Furthermore, a BCH type bound is given for free one generator skew quasi cyclic codes. We investigate the dual of skew quasi cyclic codes over R. We give a necessary and sufficient condition for skew cyclic codes over R to contain its dual. Moreover, we construct quantum codes from skew cyclic codes over R. By using computer search we give some examples about skew quasi cyclic codes and list some quantum parameters in the table. [ABSTRACT FROM AUTHOR]

We introduce a new class of evaluation linear codes by evaluating polynomials at the roots of a suitable trace function. We give conditions for self-orthogonality of these codes and their subfield-subcodes with respect to the Hermitian inner product. They allow us to construct stabilizer quantum codes over several finite fields which substantially improve the codes in the literature. For the binary case, we obtain records at http://codetables.de/. Moreover, we obtain several classical linear codes over the field $\mathbb {F}_{4}$ which are records at http://codetables.de/. [ABSTRACT FROM AUTHOR]

Let R=Fq+v1Fq+⋯+vrFq, where q is a power of a prime, vi2=vi,vivj=vjvi=0 for 1≤i,j≤r and r≥1. In this paper, the structure of cyclic codes over the ring R is studied and a Gray map ϕ from Rn to Fq(r+1)n is given. We give a construction of quantum codes from cyclic codes over the ring R. We derive Euclidean dual containing codes over Fq and Hermitian dual containing codes over Fp2m as Gray images of cyclic codes over R. In particular, we use r+1 codes associated with a cyclic code over R of arbitrary length to determine the parameters of the corresponding quantum code. Furthermore, some new non-binary quantum codes are obtained. [ABSTRACT FROM AUTHOR]

In this paper, quantum codes over Fp from cyclic codes over the ring Fp[u, v]/〈u2 − 1, v3 − v, uv − vu〉, where u2 = 1, v3 = v, uv = vu and p is an odd prime have been studied. We give the structure of cyclic codes over the ring Fp[u, v]/〈u2 − 1, v3 − v, uv − vu〉 and obtain quantum codes over Fp using self-orthogonal property of these classes of codes. Moreover, by using decomposing method, the parameters of the associated quantum code have been determined. [ABSTRACT FROM AUTHOR]

Classical Bose-Chaudhuri-Hocquenghem (BCH) codes over finite fields have been studied extensively. The Calderbank-Shor-Steane (CSS) construction, especially Steane's enlargement, and Hermitian construction are the most widely used methods in design of quantum codes. The BCH codes containing their Euclidean dual or Hermitian dual codes can be used to generate good stabilizer codes. Therefore, we can construct quantum codes by classical BCH codes over finite fields in this paper. Firstly, we study the properties of such classical BCH codes in terms of the cyclotomic cosets. It is convenient to compute the dimension of new quantum BCH codes. Meanwhile, it ensures that classical BCH codes are Euclidean dual-containing or Hermitian dual-containing. These results about suitable cyclotomic cosets make it possible to construct several new families of nonbinary quantum BCH codes with a given parameter set. Compared with the ones available in the literature, the quantum BCH codes in our schemes have good parameters. In particular, we extend to more general cases than known results. [ABSTRACT FROM AUTHOR]

In this paper, we study the structural properties of ( α + u 1 β + u 2 γ + u 1 u 2 δ ) -constacyclic codes over R = F q [ u 1 , u 2 ] / 〈 u 1 2 − u 1 , u 2 2 − u 2 , u 1 u 2 − u 2 u 1 〉 where q = p m for odd prime p and m ≥ 1 . We derive the generators of constacyclic and dual constacyclic codes. We have shown that Gray image of a constacyclic code of length n is a quasi constacyclic code of length 4 n . Also we have classified all possible self dual linear codes over this ring R . We have given the applications by computing non-binary quantum codes over this ring R .

Abstract There are some differences between quantum and classical error corrections [Nielsen M.A., and I.L. Chuang, "Quantum Computation and Quantum Information", Cambridge University Press, Cambridge, 2002.]. 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 [Huber, K., " Codes over Eisenstein-Jacobi integers ", Contemporary Mathematics 168 (1994), 165.]. There is an efficient algorithm for the encoding and decoding procedures of these codes [Huber, K., " Codes over Eisenstein-Jacobi integers ", Contemporary Mathematics 168 (1994), 165.]. 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 [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.]. 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. [ABSTRACT FROM AUTHOR]

Let q be an odd prime power and m be a positive integer. Maximum designed distance such that negacyclic BCH codes over 𝔽 q 2 of length n = q 2 m − 1 2 are Hermitian dual-containing codes is given. The dimension of such Hermitian dual-containing negacyclic codes is completely determined by analyzing cyclotomic cosets. Quantum negacyclic BCH codes of length n = q 2 m − 1 2 are obtained by using Hermitian construction. The constructed quantum negacyclic BCH codes produce new quantum codes with parameters better than those obtained from quantum BCH codes. [ABSTRACT FROM AUTHOR]

In this paper, structural properties of ( 1 − 2 v ) -constacyclic codes over the finite non-chain ring 𝔽 q + u 𝔽 q + v 𝔽 q + u v 𝔽 q are studied, where u 2 = u , v 2 = v , u v = v u and q is a power of some odd prime. As an application, some better quantum codes, compared with previous work, are obtained. [ABSTRACT FROM AUTHOR]

We give sufficient conditions for self-orthogonality with respect to symplectic, Euclidean and Hermitian inner products of a wide family of quasi-cyclic codes of index two. We provide lower bounds for the symplectic weight and the minimum distance of the involved codes. Supported in the previous results, we show algebraic constructions of good quantum codes and determine their parameters. [ABSTRACT FROM AUTHOR]

Quantum error-correcting codes are studied from classical matrix product codes point of view. Two methods to construct quantum codes from matrix product codes are provided. These constructions are applied to obtain numerous new quantum codes, some of them have better parameters than current quantum codes available. [ABSTRACT FROM AUTHOR]

In this paper, we study linear and cyclic codes over the ring F2+uF2+vF2. The ring F2+uF2+vF2 is the smallest non-Frobenius ring. We characterize the structure of cyclic codes over the ring R=F2+uF2+vF2 using of the work Abualrub and Saip (Des Codes Cryptogr 42:273-287, 2007). We study the rank and dual of cyclic codes of odd length over this ring. Specially, we show that the equation |C||C⊥|=|R|n does not hold in general for a cyclic code C of length n over this ring. We also obtain some optimal binary codes as the images of cyclic codes over the ring F2+uF2+vF2 under a Gray map, which maps Lee weights to Hamming weights. Finally, we give a condition for cyclic codes over R that contains its dual and find quantum codes over F2 from cyclic codes over the ring F2+uF2+vF2. [ABSTRACT FROM AUTHOR]

Traditional quantum error-correcting codes are designed for the depolarizing channel modeled by generalized Pauli errors occurring with equal probability. Amplitude damping channels model, in general, the decay process of a multilevel atom or energy dissipation of a bosonic system with Markovian bath at zero temperature. We discuss quantum error-correcting codes adapted to amplitude damping channels for higher dimensional systems (qudits). For multi-level atoms, we consider a natural kind of decay process, and for bosonic systems, we consider the qudit amplitude damping channel obtained by truncating the Fock basis of the bosonic modes (e.g., the number of photons) to a certain maximum occupation number. We construct families of single-error-correcting quantum codes that can be used for both cases. Our codes have larger code dimensions than the previously known single-error-correcting codes of the same lengths. In addition, we present families of multi-error correcting codes for these two channels, as well as generalizations of our construction technique to error-correcting codes for the qutrit $V$ and $\Lambda $ channels. [ABSTRACT FROM AUTHOR]

In this paper, we study cyclic codes over R=𝔽4+u𝔽4,u2=0. A necessary and sufficient condition for a cyclic code over R to contain its dual is determined. The odd and even length cases are discussed separately to obtain above condition. It is shown that Gray image of a cyclic code over R containing its dual is a linear code over 𝔽4 which also contains its dual. We have then obtained the parameters of corresponding CSS-quantum codes over 𝔽4. By augmentation, we construct codes with dual-containing property from codes of smaller size containing their duals. Through this construction, we have obtained some optimal quantum codes over 𝔽4. Some examples have been given to illustrate the results. [ABSTRACT FROM AUTHOR]

Recently, La Guardia constructed some new quantum codes from cyclic codes (La Guardia, Int. J. Theor. Phys., 2017). Inspired by this work, we consider quantum codes construction from negacyclic codes, not equivalent to cyclic codes, with only one cyclotomic coset containing at least two odd consecutive integers of even length. Some new quantum codes are obtained by this class of negacyclic codes. [ABSTRACT FROM AUTHOR]

In the paper, we consider a finite non-chain commutative ring , where . We mainly study the construction of quantum codes from cyclic codes over . For this, we obtained self-orthogonal codes over as gray images of linear and cyclic codes over , then these codes over correspond to a cyclic code over of odd length used to determine the parameters of the quantum codes. [ABSTRACT FROM AUTHOR]

In this paper, we construct quantum synchronizable codes (QSCs) based on the sum and intersection of cyclic codes. Further, infinite families of QSCs are obtained from BCH and duadic codes. Moreover, we show that the work of Fujiwara (Phys. Rev. A 87(02): 23-44, 2013) can be generalized to repeated root cyclic codes such that QSCs are always obtained, which is not the case with simple root cyclic codes. The usefulness of this extension is illustrated via examples of infinite families of QSCs from repeated root duadic codes. Finally, QSCs are constructed from the product of cyclic codes. [ABSTRACT FROM AUTHOR]

Let q be a prime power and $$m\ge 2$$ an integer. In this paper, based on classical $$q^{2}$$ -ary constacyclic codes, we apply the Hermitian construction to obtain several classes of q-ary quantum stabilizer codes of length $$(q^{2m}-1)/(q+1)$$ . These quantum codes have parameters better than those obtained from classical BCH codes. [ABSTRACT FROM AUTHOR]