Construct Quantum Codes Research Articles

New quantum codes from constacyclic codes over a general non-chain ring

Let [Formula: see text] be a prime power and let [Formula: see text] be a finite non-chain ring, where [Formula: see text], are polynomials, not all linear, which split into distinct linear factors over [Formula: see text]. We characterize constacyclic codes over the ring [Formula: see text] and study quantum codes from these. As an application, some new and better quantum codes, as compared to the best known codes, are obtained. We also prove that the choice of the polynomials [Formula: see text], [Formula: see text], is irrelevant while constructing quantum codes from constacyclic codes over [Formula: see text], it depends only on their degrees.

Constacyclic Codes Over 𝑭𝒒[𝒖] /〈𝒖𝟑 = 𝟎〉 and Their Application of Constructing Quantum Codes

Let 𝑅 = 𝐹𝑞+u𝐹𝑞+𝑢^2𝐹𝑞, 𝑢^3=0 be a finite chain ring. In this paper, we give the structure of constacyclic codes over 𝑅 and obtain self-orthogonal codes over 𝐹𝑞 by using the Gray map from 𝑅𝑛 to 𝐹𝑞^(3𝑛). As an application, we present a construction of quantum codes from the codes obtained from this class.

Constacyclic codes over $$F_q[u_{1}, u_{2}, \cdots , u_{k}]/\langle u_{i}^3-u_{i},u_{i}u_{j}-u_{j}u_{i}\rangle $$ and their applications of constructing quantum codes

Constacyclic codes over $$F_q[u_{1}, u_{2}, \cdots , u_{k}]/\langle u_{i}^3-u_{i},u_{i}u_{j}-u_{j}u_{i}\rangle $$ and their applications of constructing quantum codes

A database of quantum codes

Quantum error correcting codes (QECC) is becoming an increasingly important branch of coding theory. For classical block codes, a comprehensive database of best known codes (codetables.de) exists which is available online at \cite{codetables}. The same database contains data on best known quantum codes as well, but only for the binary field. There has been an increased interest in quantum codes over larger fields with many papers reporting such codes in the literature. However, to the best of our knowledge, there is no database of best known quantum codes for most fields. We established a new database of QECC that includes codes over $\mathbb{F}_{q^2}$ for $q\leq 29$. We also present several methods of constructing quantum codes from classical codes based on the CSS construction. We have found dozens of new quantum codes that improve the previously known parameters and also hundreds new quantum codes that did not exist in the literature.

Infinite families of $ t $-designs and strongly regular graphs from punctured codes

<p style='text-indent:20px;'>The puncturing technique is sometimes efficient in constructing projective codes from original codes which are not projective. In this paper, several families of projective linear codes punctured from reducible cyclic codes, special linear codes or irreducible cyclic codes are investigated. The parameters and weight enumerators of the punctured codes and their duals are explicitly determined. Some of the codes are optimal and some of the codes are self-orthogonal which can be used to construct quantum codes. Several infinite families of combinatorial <inline-formula><tex-math id="M2">\begin{document}$ 2 $\end{document}</tex-math></inline-formula>-designs and <inline-formula><tex-math id="M3">\begin{document}$ 3 $\end{document}</tex-math></inline-formula>-designs including some families of Steiner systems are constructed from the punctured codes and their duals. Besides, infinite families of strongly regular graphs are also derived from some families of two-weight projective codes.</p>

AG codes from {{mathbb{F}}_{q^7}}-rational points of the GK maximal curve

In Beelen and Montanucci (Finite Fields Appl 52:10–29, 2018) and Giulietti and Korchmáros (Math Ann 343:229–245, 2009), Weierstrass semigroups at points of the Giulietti–Korchmáros curve {mathcal {X}} were investigated and the sets of minimal generators were determined for all points in {mathcal {X}}(mathbb {F}_{q^2}) and {mathcal {X}}(mathbb {F}_{q^6})setminus {mathcal {X}}( mathbb {F}_{q^2}). This paper completes their work by settling the remaining cases, that is, for points in {mathcal {X}}(overline{mathbb {F}}_{q}){setminus }{mathcal {X}}( mathbb {F}_{q^6}). As an application to AG codes, we determine the dimensions and the lengths of duals of one-point codes from a point in {mathcal {X}}(mathbb {F}_{q^7}){setminus }{mathcal {X}}( mathbb {F}_{q}) and we give a bound on the Feng–Rao minimum distance d_{ORD}. For q=3 we provide a table that also reports the exact values of d_{ORD}. As a further application we construct quantum codes from mathbb {F}_{q^7}-rational points of the GK-curve.

Some New Quantum BCH Codes over Finite Fields.

Quantum error correcting codes (QECCs) play an important role in preventing quantum information decoherence. Good quantum stabilizer codes were constructed by classical error correcting codes. In this paper, Bose–Chaudhuri–Hocquenghem (BCH) codes over finite fields are used to construct quantum codes. First, we try to find such classical BCH codes, which contain their dual codes, by studying the suitable cyclotomic cosets. Then, we construct nonbinary quantum BCH codes with given parameter sets. Finally, a new family of quantum BCH codes can be realized by Steane’s enlargement of nonbinary Calderbank-Shor-Steane (CSS) construction and Hermitian construction. We have proven that the cyclotomic cosets are good tools to study quantum BCH codes. The defining sets contain the highest numbers of consecutive integers. Compared with the results in the references, the new quantum BCH codes have better code parameters without restrictions and better lower bounds on minimum distances. What is more, the new quantum codes can be constructed over any finite fields, which enlarges the range of quantum BCH codes.

A class of skew cyclic codes and application in quantum codes construction

Let p be an odd prime, and k an integer such that k∣(p−1). This paper studies quantum codes from skew cyclic codes over the ring R=Fq[u]〈uk+1−u〉, where q is a power of the prime p. We construct a set of orthogonal idempotents of the ring R and using the set, skew cyclic codes over the ring R are decomposed as direct sum of skew cyclic codes over Fq. We obtain a necessary and sufficient condition for a skew cyclic code to contain its dual. As an application, we construct quantum codes from skew cyclic codes over Fq. It is observed that some quantum codes we constructed are MDS quantum codes.

On some binary symplectic self-orthogonal codes

Symplectic self-orthogonal codes over finite fields are an important class of linear codes in coding theory, which can be used to construct quantum codes. In this paper, characterizations of symplectic self-orthogonal codes over finite fields $$F_{q}$$ are given. A necessary and sufficient condition for determining symplectic self-orthogonal codes is obtained. Several classes of symplectic self-orthogonal codes are constructed. Furthermore, the symplectic weight distributions of some new classes of binary symplectic self-orthogonal codes are completely determined.

Constructions of quasi-twisted quantum codes

In this work, our main objective is to construct quantum codes from quasi-twisted (QT) codes. At first, a necessary and sufficient condition for Hermitian self-orthogonality of QT codes is introduced by virtue of the Chinese remainder theorem. Then, we utilize these self-orthogonal QT codes to provide quantum codes via the famous Hermitian construction. Moreover, we present a new construction method of q-ary quantum codes, which can be viewed as an effective generalization of the Hermitian construction. General QT codes that are not self-orthogonal are also employed to construct quantum codes. As the computational results, some binary, ternary and quaternary quantum codes are constructed and their parameters are determined, which all cannot be deduced by the quantum Gilbert–Varshamov bound. In the binary case, a small number of quantum codes are derived with strictly improved parameters compared with the current records. In the ternary and quaternary cases, our codes fill some gaps or have better performances than the current results.

Quantum bicyclic hyperbolic codes

Bicyclic codes are a generalization of the one-dimensional (1D) cyclic codes to two dimensions (2D). Similar to the 1D case, in some cases, 2D cyclic codes can also be constructed to guarantee a specified minimum distance. Many aspects of these codes are yet unexplored. Motivated by the problem of constructing quantum codes, we study some structural properties of certain bicyclic codes. We show that a primitive narrow-sense bicyclic hyperbolic code of length \(n^2\) contains its dual if and only if its design distance is lower than \(n-O(\sqrt{n})\). We extend the sufficiency condition to the non-primitive case as well. We also show that over quadratic extension fields, a primitive bicyclic hyperbolic code of length \(n^2\) contains Hermitian dual if and only if its design distance is lower than \(n-O(\sqrt{n})\). Our results are analogous to some structural results known for BCH and Reed–Solomon codes. They further our understanding of bicyclic codes. We also give an application of these results by showing that we can construct two classes of quantum bicyclic codes based on our results.

$${\mathbb {F}}_qR$$-linear skew constacyclic codes and their application of constructing quantum codes

Let q be a prime power with $$\mathrm{gcd}(q,6)=1$$ . Let $$R={\mathbb {F}}_{q^2}+u{\mathbb {F}}_{q^2}+v{\mathbb {F}}_{q^2}+uv{\mathbb {F}}_{q^2}$$ , where $$u^2=u$$ , $$v^2=v$$ and $$uv=vu$$ . In this paper, we give the definition of linear skew constacyclic codes over $${\mathbb {F}}_{q^2}R$$ . By the decomposition method, we study the structural properties and determine the generator polynomials and the minimal generating sets of linear skew constacyclic codes. We define a Gray map from $${\mathbb {F}}_{q^2}^{\alpha }\times R^{\beta }$$ to $${\mathbb {F}}_{q^2}^{\alpha +4\beta }$$ preserving the Hermitian orthogonality, where $$\alpha $$ and $$\beta $$ are positive integers. As an application, by Hermitian construction, we obtain some good quantum error-correcting codes.

A Study of FqR-Cyclic Codes and Their Applications in Constructing Quantum Codes

Let R = F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> + uF <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> + vF <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> + uvF <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> , with u <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> = u, v <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> = v, uv = vu, where q = p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> for a positive integer m and an odd prime p. We study the algebraic structure of F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> R-cyclic codes of block length (r, s). These codes can be viewed as R[x]-submodules of F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> [x]/(x <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sup> - 1) × R[x]/(x <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sup> - 1). For this family of codes we discuss the generator polynomials and minimal generating sets. We study the algebraic structure of separable codes. Further, we discuss the duality of this family of codes and determine their generator polynomials. We obtain several optimal and near-optimal codes from this study. As applications, we discuss a construction of quantum error-correcting codes (QECCs) from F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> R-cyclic codes and construct some good QECCs.

Modifying Method of Constructing Quantum Codes From Highly Entangled States

There is a connection between classical codes, highly entangled pure states (called k-uniform or absolutely maximally entangled (AME) states), and quantum error correcting codes (QECCs). This leads to a systematic method to construct stabilizer QECCs by starting from a k-uniform state or the corresponding classical code and tracing out one party at each step. We provide explicit constructions for codewords, encoding procedure and stabilizer formalism of the QECCs by describing the changes that partial traces cause on the corresponding generator matrix of the classical codes. We then modify the method to produce another set of stabilizer QECCs that encode a logical qudit into a subspace spanned by AME states. This construction produces quantum codes starting from an AME state without tracing out any party. Therefore, quantum stabilizer codes with larger codespace can be constructed.

New Non-Binary Quantum Codes from Cyclic Codes Over Product Rings

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.

Hermitian dual-containing narrow-sense constacyclic BCH codes and quantum codes

Cyclic codes are an interesting class of linear codes due to their efficient encoding and decoding algorithms. Bose–Chaudhuri–Hocquenghem (BCH) codes which form a significant subclass of cyclic codes are important in both theory and practice since they have good error-correcting capabilities and have been widely used in communication systems, storage devices, and so on. Quantum codes with good parameters can be also constructed from BCH codes. In this paper, we construct q-ary quantum codes of length $$\frac{q^{2m}-1}{\rho }$$ using constacyclic BCH codes with order $$\rho $$ and cyclic BCH codes, respectively, where $$\rho $$ divides $$q+1$$ , q is a prime power and m is a positive integer. By comparing the obtained quantum codes, we get that constacyclic BCH codes are a better resource in constructing quantum codes than cyclic BCH codes in general. Compared with the quantum codes available in Aly et al. (IEEE Trans Inf Theory 53(3): 1183–1188, 2007) and Zhang et al. (IEEE Access 4:36122, 2018), the quantum codes in our schemes have better parameters. In particular, we extend some known results in Kai et al. (Int J Quantum Inf 16(7):1850059, 2018), La Guardia (Phys Rev A 80(4):042331, 2009), Li et al. (Quantum Inf Comput 12:0021–0035, 2013), Lin (IEEE Trans Inf Theory 50(3):5551–5554, 2004), Tang et al. (IEICE Trans Fund E102-A(1):303–306, 2019), Wang and Zhu (Quantum Inf Process 14(3):881–889, 2015), Yuan et al. (Des Codes Cryptogr 85(1):179–190, 2017) to more general case.

Optimal quantum subsystem codes in two dimensions

Given any two classical codes with parameters $[n_1,k,d_1]$ and $[n_2,k,d_2]$, we show how to construct a quantum subsystem code in 2-dimensions with parameters $[[N,K,D]]$ satisfying $N\le 2n_1n_2$, $K=k$, and $D=\min(d_1,d_2)$. These quantum codes are in the class of generalized Bacon-Shor codes introduced by Bravyi. We note that constructions of good classical codes can be used to construct quantum codes that saturate Bravyi's bound $KD=O(N)$ on the code parameters of 2-dimensional subsystem codes. One of these good constructions uses classical expander codes. This construction has the additional advantage of a linear time quantum decoder based on the classical Sipser-Spielman flip decoder. Finally, while the subsystem codes we create do not have asymptotic thresholds, we show how they can be gauge-fixed to certain hypergraph product codes that do.

New constructions of entanglement-assisted quantum MDS codes from negacyclic codes

When constructing quantum codes under the entanglement-assisted (EA) stabilizer formalism, one can ignore the limitation of dual-containing condition. This allows us to construct EA quantum error-correcting codes (QECCs) from any classical linear codes. The main contribution of this manuscript is to make a general statement for determining the optimal number of pre-shared qubits instead of presenting only specific cases. Let [Formula: see text] and [Formula: see text], where [Formula: see text] is an odd prime power, [Formula: see text] and [Formula: see text]. By deeply investigating the decomposition of the defining set of negacyclic codes, we generalize the number of pre-shared entanglement pairs of Construction (1) in Lu et al. [Quantom Inf. Process. 17 (2018) 69] from [Formula: see text] to arbitrary even numbers less than or equal to [Formula: see text]. Consequently, a series of EA quantum maximum distance separable (EAQMDS) codes can be produced. The absolute majority of them are new and the minimum distance can be up to [Formula: see text]. Moreover, this method can be applied to construct many other families of EAQECCs with good parameters, especially large minimum distance.

Skew quasi cyclic codes over 𝔽q + v𝔽q

In this work, skew quasi cyclic codes over [Formula: see text], where [Formula: see text] are considered. The generating set for one generator skew quasi cyclic codes over [Formula: see text] 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 [Formula: see text]. We give a necessary and sufficient condition for skew cyclic codes over [Formula: see text] to contain its dual. Moreover, we construct quantum codes from skew cyclic codes over [Formula: see text]. By using computer search we give some examples about skew quasi cyclic codes and list some quantum parameters in the table.

Factorizations of Binomial Polynomials and Enumerations of LCD and Self-Dual Constacyclic Codes

Constacyclic codes are well-known generalizations of cyclic and negacyclic codes. Due to their rich algebraic structure, constacyclic codes are used to construct quantum codes and symbol-pair codes. Let <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathbb {F}}_{q}$ </tex-math></inline-formula> be a finite field with order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> is a positive power of a prime <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> . Suppose that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> is a positive integer and the product of distinct prime factors of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> divides <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q-1$ </tex-math></inline-formula> , i.e., <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$rad(n)\mid (q-1)$ </tex-math></inline-formula> . In this paper, we explicitly factorize the polynomial <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$x^{n}-\lambda $ </tex-math></inline-formula> for each <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\lambda \in {\mathbb {F}}_{q}^{*}$ </tex-math></inline-formula> . As applications, first, we obtain all repeated-root <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\lambda $ </tex-math></inline-formula> -constacyclic codes and their dual codes of length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$np^{s}$ </tex-math></inline-formula> over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathbb {F}}_{q}$ </tex-math></inline-formula> ; second, we determine all simple-root LCD cyclic codes and LCD negacyclic codes of length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathbb {F}}_{q}$ </tex-math></inline-formula> ; third, we list all self-dual repeated-root negacyclic codes of length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$np^{s}$ </tex-math></inline-formula> over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathbb {F}}_{q}$ </tex-math></inline-formula> . In contrast to known results, the lengths of constacyclic codes in this paper have more flexible parameters.