Hermitian Construction Research Articles

Two classes of quantum codes from almost MDS codes

Hermitian dual-containing codes are an important class of linear codes which have important applications in the construction of quantum codes. In this paper, two classes of Hermitian dual-containing almost MDS codes over finite fields are studied. By employing the Hermitian construction, a class of quantum codes with minimum distance [Formula: see text] and a class of quantum codes with minimum distance [Formula: see text] are constructed.

Hermitian and Unitary Almost-Companion Matrices of Polynomials on Demand

We introduce the concept of the almost-companion matrix (ACM) by relaxing the non-derogatory property of the standard companion matrix (CM). That is, we define an ACM as a matrix whose characteristic polynomial coincides with a given monic and generally complex polynomial. The greater flexibility inherent in the ACM concept, compared to CM, allows the construction of ACMs that have convenient matrix structures satisfying desired additional conditions, compatibly with specific properties of the polynomial coefficients. We demonstrate the construction of Hermitian and unitary ACMs starting from appropriate third-degree polynomials, with implications for their use in physical-mathematical problems, such as the parameterization of the Hamiltonian, density, or evolution matrix of a qutrit. We show that the ACM provides a means of identifying the properties of a given polynomial and finding its roots. For example, we describe the ACM-based solution of cubic complex algebraic equations without resorting to the use of the Cardano-Dal Ferro formulas. We also show the necessary and sufficient conditions on the coefficients of a polynomial for it to represent the characteristic polynomial of a unitary ACM. The presented approach can be generalized to complex polynomials of higher degrees.

New quantum codes derived from the images of constacyclic codes

Assume that $q$ is a prime power and $m\geq 2$ is a positive integer. Cyclic codes over $\mathbb{F}_{q^{2m}}$ of length $n=\frac{q^{2m}-1}{\rho }$ with $\rho\mid (q-1)$, and constacyclic codes over $\mathbb{F}_{q^{2m}}$ of length $n=\frac{q^{2m}-1}{\rho }$ with $\rho\mid (q+1)$ are considered in this paper, respectively. Two classes of quantum codes are derived from the images of these codes by the Hermitian construction. Compared with the previously known quantum codes, the quantum codes in our scheme have better parameters.

Entanglement-Assisted Quantum Codes from Cyclic Codes.

Entanglement-assisted quantum-error-correcting (EAQEC) codes are quantum codes which use entanglement as a resource. These codes can provide better error correction than the (entanglement unassisted) codes derived from the traditional stabilizer formalism. In this paper, we provide a general method to construct EAQEC codes from cyclic codes. Afterwards, the method is applied to Reed-Solomon codes, BCH codes, and general cyclic codes. We use the Euclidean and Hermitian construction of EAQEC codes. Three families have been created: two families of EAQEC codes are maximal distance separable (MDS), and one is almost MDS or almost near MDS. The comparison of the codes in this paper is mostly based on the quantum Singleton bound.

On the complete weight distributions of quantum error-correcting codes

In a recent paper, Hu et al. defined the complete weight distributions of quantum codes and proved the MacWilliams identities, and as applications they showed how such weight distributions may be used to obtain the singleton-type and hamming-type bounds for asymmetric quantum codes. In this paper we extend their study much further and obtain several new results concerning the complete weight distributions of quantum codes and applications. In particular, we provide a new proof of the MacWilliams identities of the complete weight distributions of quantum codes. We obtain new information about the weight distributions of quantum MDS codes and the double weight distribution of asymmetric quantum MDS codes. We get new identities involving the complete weight distributions of two different quantum codes. We estimate the complete weight distributions of quantum codes under special conditions and show that quantum BCH codes by the Hermitian construction from primitive, narrow-sense BCH codes satisfy these conditions and hence these estimate applies.

On construction of quantum codes with dual-containing quasi-cyclic codes

One of the main objectives of quantum error-correction theory is to construct quantum codes with optimal parameters and properties. In this paper, we propose a class of 2-generator quasi-cyclic codes and study their applications in the construction of quantum codes over small fields. Firstly, some sufficient conditions for these 2-generator quasi-cyclic codes to be dual-containing concerning Hermitian inner product are determined. Then, we utilize these Hermitian dual-containing quasi-cyclic codes to produce quantum codes via the famous Hermitian construction. Moreover, we present a lower bound on the minimum distance of these quasi-cyclic codes, which is helpful to construct quantum codes with larger lengths and dimensions. As the computational results, many new quantum codes that exceed the quantum Gilbert-Varshamov bound are constructed over $F_q$, where $q$ is $2,3,4,5$. In particular, 16 binary quantum codes raise the lower bound on the minimum distance in Grassl's table \cite{Grassl:codetables}. In nonbinary cases, many quantum codes are new or have better parameters than those in the literature.

Entanglement-Assisted Quantum Codes From Algebraic Geometry Codes

Quantum error-correcting codes play the role of suppressing noise and decoherence in quantum systems by introducing redundancy. Some strategies can be used to improve the parameters of these codes. For example, entanglement can provide a way for quantum error-correcting codes to achieve higher rates than the one obtained by means of the traditional stabilizer formalism. Such codes are called entanglement-assisted quantum error-correcting (EAQEC) codes. In this paper, we utilize algebraic geometry codes to construct several families of EAQEC codes derived from the Euclidean and the Hermitian construction. Three families constructed here consist of codes whose quantum Singleton defect is equal to zero, one, or two. We also construct families of EAQEC codes with an encoding rate exceeding the quantum Gilbert-Varshamov bound. Additionally, asymptotically good towers of linear complementary dual codes are used to obtain asymptotically good families of EAQEC codes consuming maximal entanglement. Furthermore, a simple comparison with the quantum Gilbert-Varshamov bound demonstrates that, by utilizing the proposed construction, it is possible to generate an asymptotically family of EAQEC codes that exceeds this bound.

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.

Quantum codes from Hermitian dual-containing constacyclic codes over $${\mathbb {F}}_{q^{2}}+{v}{\mathbb {F}}_{q^{2}}$$

Let $${\mathbb {R}}$$ be the finite non-chain ring $${\mathbb {F}}_{{ q}^{2}}+{v}{\mathbb {F}}_{{ q}^{2}}$$ , where $${v}^{2}={v}$$ and q is an odd prime power. In this paper, we study quantum codes over $${\mathbb {F}}_{{ q}}$$ from constacyclic codes over $${\mathbb {R}}$$ . We define a class of Gray maps, which preserves the Hermitian dual-containing property of linear codes from $${\mathbb {R}}$$ to $${\mathbb {F}}_{{ q}^{2}}$$ . We study $${\alpha }(1-2v)$$ -constacyclic codes over $${\mathbb {R}}$$ , and show that the images of $$\alpha (1-2v)$$ -constacyclic codes over $${\mathbb {R}}$$ under the special Gray map are $$\alpha ^{2}$$ -constacyclic codes over $${\mathbb {F}}_{{ q}^{2}}$$ . Some new non-binary quantum codes are obtained via the Gray map and the Hermitian construction from Hermitian dual-containing $$\alpha (1-2v)$$ -constacyclic codes over $${\mathbb {R}}$$ .

Construction of new quantum codes via Hermitian dual-containing matrix-product codes

In 2001, Blackmore and Norton introduced an important tool called matrix-product codes, which turn out to be very useful to construct new quantum codes of large lengths. To obtain new and good quantum codes, we first give a general approach to construct matrix-product codes being Hermitian dual-containing and then provide the constructions of such codes in the case $$s{\mid }(q^{2}-1)$$ , where s is the number of the constituent codes in a matrix-product code. For $$s{\mid } (q+1)$$ , we construct such codes with lengths more flexible than the known ones in the literature. For $$s{\mid } (q^{2}-1)$$ and $$s{\not \mid } (q+1)$$ , such codes are constructed in an unusual manner; some of the constituent codes therein are not required to be Hermitian dual-containing. Accordingly, by Hermitian construction, we present two procedures for acquiring quantum codes. Finally, we list some good quantum codes, many of which improve those available in the literature or add new parameters.

MDS Constacyclic Codes of Prime Power Lengths Over Finite Fields and Construction of Quantum MDS Codes

If we fix the code length n and dimension k, maximum distance separable (briefly, MDS) codes form an important class of codes because the class of MDS codes has the greatest error-correcting and detecting capabilities. In this paper, we establish all MDS constacyclic codes of length ps over $\mathbb {F}_{p^{m}}$ . We also give some examples of MDS constacyclic codes over finite fields. As an application, we construct all quantum MDS codes from repeated-root codes of prime power lengths over finite fields using the CSS and Hermitian constructions. We provide all quantum MDS codes constructed from dual codes of repeated-root codes of prime power lengths over finite fields using the Hermitian construction. They are new in the sense that their parameters are different from all the previous constructions. Moreover, some of them have larger Hamming distances than the well known quantum error-correcting codes in the literature.

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.

New quantum codes from matrix-product codes over small fields

In this paper, we provide methods for constructing Hermitian dual-containing (HDC) matrix-product codes over $$\mathbb {F}_{q^2}$$ from some non-singular matrices and a special sequence of HDC codes and determine parameters of obtained matrix-product codes when the input matrix and sequence of HDC codes satisfy some conditions. Then, using some nested HDC BCH codes with lengths $$n=\frac{q^4-1}{a} (a=1 ~$$ or $$~ a=q\pm 1)$$ , we construct some HDC matrix-product codes with lengths $$N=$$ 2n or 3n and derive nonbinary quantum codes with length N from these matrix-product codes via Hermitian construction. Four classes of quantum codes over $$\mathbb {F}_{q}$$ ( $$3\le q\le 5$$ ) are presented, whose parameters are better than those in the literature. Besides, some of our new quantum codes can exceed the quantum Gilbert-Varshamov (GV) bound.

The images of constacyclic codes and new quantum codes

Let q be a prime power and $$m\ge 2$$ be a positive integer. A sufficient condition for the $$q^2$$-ary images of constacyclic codes over $${\mathbb {F}}_{q^{2m}}$$ to be Hermitian self-orthogonal is presented. Hermitian self-orthogonal codes over $${\mathbb {F}}_{q^{2}}$$ are obtained as the images of constacyclic codes over $${\mathbb {F}}_{q^{2m}}$$. Two classes of quantum codes are derived by employing the Hermitian construction. The construction produces quantum codes with better parameters than the previously known ones.

$${\mathbb {F}}_qR$$-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.

Nonbinary quantum codes from constacyclic codes over polynomial residue rings

Let R be the polynomial residue ring $${\mathbb {F}}_{q^{2}}+u{\mathbb {F}}_{q^{2}}$$ , where $${\mathbb {F}}_{q^2}$$ is the finite field with $$q^2$$ elements, q is a power of a prime p, and u is an indeterminate with $$u^{2}=0.$$ We introduce a Gray map from R to $${\mathbb {F}}_{q^{2}}^{p}$$ and study $$(1-u)$$-constacyclic codes over R. It is proved that the image of a $$(1-u)$$-constacyclic code of length n over R under the Gray map is a distance-invariant linear cyclic code of length pn over $${\mathbb {F}}_{q^{2}}.$$ We give some necessary and sufficient conditions for $$(1-u)$$-constacyclic codes over R to be Hermitian dual-containing. In particular, a new class of $$2^{m}$$-ary quantum codes is obtained via the Gray map and the Hermitian construction from Hermitian dual-containing $$(1-u)$$-constacyclic codes over the ring $${\mathbb {F}}_{2^{2m}}+u{\mathbb {F}}_{2^{2m}}$$.

Construction of new quantum codes derived from constacyclic codes over $${\mathbb {F}}_{q^2}+u{\mathbb {F}}_{q^2}+\cdots +u^{r-1}{\mathbb {F}}_{q^2}$$

Let $$R={\mathbb {F}}_{q^2}+u{\mathbb {F}}_{q^2}+\cdots +u^{r-1}{\mathbb {F}}_{q^2}$$ be a finite non-chain ring, where q is a prime power, $$u^{r}=1$$ and $$r|(q+1)$$. In this paper, we study u-constacyclic codes over the ring R. Using the matrix of Fourier transform, a Gray map from R to $${\mathbb {F}}_{q^2}^{r}$$ is given. Under the special Gray map, we show that the image of Gray map of u-constacyclic codes over R are cyclic codes over $${\mathbb {F}}_{q^2}$$, and some new quantum codes are obtained via the Gray map and Hermitian construction from Hermitian dual-containing u-constacyclic codes.

Some negacyclic BCH codes and quantum codes

In this paper, we investigate narrow-sense and non-narrow-sense negacyclic Bose–Chaudhuri–Hocquenghem (NBCH) codes of length $$n=\frac{q^m-1}{a}(q^m+1)$$ over $${\mathbb {F}}_{q^2}$$ closely, where q is an odd prime power, $$m\ge 3$$ is an odd integer and $$a\mid (q^m-1)$$ is an even integer. To derive accurate maximum designed distance of Hermitian dual containing NBCH codes, we define $$2\le a\le 2q^2-q-1$$ for narrow-sense codes with $$\delta _{m, a}^N$$ and $$2\le a< 2(q-1)$$ for non-narrow-sense codes with $$\delta _{m, a}^{NN}$$. For given a, our maximum designed distance improves over the distance $$\delta _m^A$$ of Aly et al. (IEEE Trans Inf Theory 53:1183–1188, 2007) to a great extent, that is, $$\delta _{m, a}^{N}=\delta _{m, a}^{NN}=\frac{a+2}{2}\delta _m^A$$. After determining dimensions of such Hermitian dual containing NBCH codes, we construct many new quantum codes via Hermitian construction naturally, whose parameters are better than the ones in the literature.

Some New Constructions of Quantum MDS Codes

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 q/2+1 Three of these six classes of quantum MDS codes have longer lengths than the ones constructed in [1] and [2], 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.

Some Nonprimitive BCH Codes and Related Quantum Codes

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.