Quantum Codes Derived from Negacyclic Codes

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.     

Quantum Codes Derived from Cyclic Codes

In this note, we present a construction of new nonbinary quantum codes with good parameters. These codes are obtained by applying the Calderbank-Shor-Steane (CSS) construction. In order to do this, we show the existence of (classical) cyclic codes whose defining set consists of only one cyclotomic coset containing at least two consecutive integers.     

A Construction of Quantum Error-Locating Codes

We present the construction of quantum error-locating(QEL) codes based on classical error-locating(EL)codes. Similar to classical EL codes, QEL codes lie midway between quantum error-correcting codes and quantum errordetecting codes. Then QEL codes can locate qubit errors within one sub-block of the received qubit symbols but do not need to determine the exact locations of the erroneous qubits. We show that, an e-error-locating code derived from an arbitrary binary cyclic code with generator polynomial g(x), can lead to a QEL code with e error-locating abilities, only if g(x) does not contain the(1 + x)-factor.     

Error Probability Mitigation in Quantum Reading Using Classical Codes

A general framework describing the statistical discrimination of an ensemble of quantum channels is given by the name quantum reading. Several tools can be applied in quantum reading to reduce the error probability in distinguishing the ensemble of channels. Classical and quantum codes can be envisioned for this goal. The aim of this paper is to present a simple but fruitful protocol for this task using classical error-correcting codes. Three families of codes are considered: Reed–Solomon codes, BCH codes, and Reed–Muller codes. In conjunction with the use of codes, we also analyze the role of the receiver. In particular, heterodyne and Dolinar receivers are taken into consideration. The encoding and measurement schemes are connected by the probing step. As probes, we consider coherent states. In such a simple manner, interesting results are obtained. As we show, there is a threshold below which using codes surpass optimal and sophisticated schemes for any fixed rate and code. BCH codes in conjunction with Dolinar receiver turn out to be the optimal strategy for error mitigation in quantum reading.     

New Optimal Asymmetric Quantum Codes Derived from Negacyclic Codes

The construction of quantum maximum-distance-separable (MDS) codes have been studied by many researchers for many years. Here, by using negacyclic codes, we construct two families of asymmetric quantum codes. The first family is the asymmetric quantum codes with parameters $[[q^{2}+1,q^{2}+1-2(t+k+1),(2k+2)/(2t+2)]]_{q^{2}}$ , where 0≤t≤k≤(q?1)/2, $q \equiv1(\operatorname{mod} 4)$ , and k, t are positive integers. The second one is the asymmetric quantum codes with parameters $[[(q^{2}+1)/2,(q^{2}+1)/2-2(t+k),(2k+1)/(2t+1)]]_{q^{2}}$ , where 1≤t≤k≤(q?1)/2, and k, t are positive integers. Moreover, the constructed asymmetric quantum codes are optimal and different from the codes available in the literature.     

一种严格最佳(ν,k,1)光正交码的设计方法

介绍了严格最佳和准最佳(ν,k,1)光正交码的定义，阐述了它们与(ν,k,1)循环差集族的关系。基于Wilson均匀分布差引理和初等数论的基本理论，提出一种最佳(ν,k,1)循环差集族的构造方法，即构造定义在ν阶有限域上满足特定约束条件的k元集合。将该方法用于光正交码的设计中，可以有效地设计一些严格最佳(ν,k,1)光正交码，其中，码长ν为素数，码重k的取值为4、5和6。最后结合具体实例，给出严格最佳(ν,k,1)光正交码的计算机辅助设计方法。同其他设计方法相比，该设计方法既简单又实用，尤其对严格最佳（ν,k,1）光正交码而言，设计效率较高；随着码重k的增加，码的设计效率逐渐降低。     

Some Families of Asymmetric Quantum MDS Codes Constructed from Constacyclic Codes

Quantum maximal-distance-separable (MDS) codes that satisfy quantum Singleton bound with different lengths have been constructed by some researchers. In this paper, seven families of asymmetric quantum MDS codes are constructed by using constacyclic codes. We weaken the case of Hermitian-dual containing codes that can be applied to construct asymmetric quantum MDS codes with parameters \([[n,k,d_{z}/d_{x}]]_{q^{2}}\). These quantum codes are able to correct quantum errors with greater asymmetry. Moreover, these quantum codes constructed in this paper are different from the ones in the literature.     

Some Families of Quantum BCH Codes

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.

     

Systematic Encoding and Shortening of PAC Codes

Polarization adjusted convolutional (PAC) codes are a class of codes that combine channel polarization with convolutional coding. PAC codes are of interest for their high performance. This paper presents a systematic encoding and shortening method for PAC codes. Systematic encoding is important for lowering the bit-error rate (BER) of PAC codes. Shortening is important for adjusting the block length of PAC codes. It is shown that systematic encoding and shortening of PAC codes can be carried out in a unified framework.     

Universal Framework for Quantum Error-Correcting Codes

We present a universal framework for quantum error-correcting codes, i.e., a framework that applies to the most general quantum error-correcting codes. This framework is based on the group algebra, an algebraic notation associated with nice error bases of quantum systems. The nicest thing about this framework is that we can characterize the properties of quantum codes by the properties of the group algebra. We show how it characterizes the properties of quantum codes as well as generates some new results about quantum codes.     

List Decoding of Arıkan’s PAC Codes

Polar coding gives rise to the first explicit family of codes that provably achieve capacity with efficient encoding and decoding for a wide range of channels. However, its performance at short blocklengths under standard successive cancellation decoding is far from optimal. A well-known way to improve the performance of polar codes at short blocklengths is CRC precoding followed by successive-cancellation list decoding. This approach, along with various refinements thereof, has largely remained the state of the art in polar coding since it was introduced in 2011. Recently, Arıkan presented a new polar coding scheme, which he called polarization-adjusted convolutional (PAC) codes. At short blocklengths, such codes offer a dramatic improvement in performance as compared to CRC-aided list decoding of conventional polar codes. PAC codes are based primarily upon the following main ideas: replacing CRC codes with convolutional precoding (under appropriate rate profiling) and replacing list decoding by sequential decoding. One of our primary goals in this paper is to answer the following question: is sequential decoding essential for the superior performance of PAC codes? We show that similar performance can be achieved using list decoding when the list size L is moderately large (say, L⩾128). List decoding has distinct advantages over sequential decoding in certain scenarios, such as low-SNR regimes or situations where the worst-case complexity/latency is the primary constraint. Another objective is to provide some insights into the remarkable performance of PAC codes. We first observe that both sequential decoding and list decoding of PAC codes closely match ML decoding thereof. We then estimate the number of low weight codewords in PAC codes, and use these estimates to approximate the union bound on their performance. These results indicate that PAC codes are superior to both polar codes and Reed–Muller codes. We also consider random time-varying convolutional precoding for PAC codes, and observe that this scheme achieves the same superior performance with constraint length as low as ν=2.     

Entanglement-Assisted Quantum Negacyclic BCH Codes

International Journal of Theoretical Physics - The entanglement-assisted quantum error correcting codes (EAQECCs) are a simple and important class of quantum codes. The entanglement-assisted...     

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.     

Analysis of Blind Reconstruction of BCH Codes

In this paper, the theoretical lower-bound on the success probability of blind reconstruction of Bose–Chaudhuri–Hocquenghem (BCH) codes is derived. In particular, the blind reconstruction method of BCH codes based on the consecutive roots of generator polynomials is mainly analyzed because this method shows the best blind reconstruction performance. In order to derive a performance lower-bound, the theoretical analysis of BCH codes on the aspects of blind reconstruction is performed. Furthermore, the analysis results can be applied not only to the binary BCH codes but also to the non-binary BCH codes including Reed–Solomon (RS) codes. By comparing the derived lower-bound with the simulation results, it is confirmed that the success probability of the blind reconstruction of BCH codes based on the consecutive roots of generator polynomials is well bounded by the proposed lower-bound.     

Nested Quantum Error Correction Codes via Subgraphs

Because of its directness and simplicity, using graph is a worthy researching approach to construct quantum error correction codes. Nested graphical quantum code is a special class of stabilizer codes. In this letter, by making uses of the entanglement of several subgraphs, we proposed a novel construction method of generator matrices of graphical quantum nested codes, and families of corresponding nested graphical quantum codes.     

On the Construction of Asymmetric Quantum Codes

Several families of good nonbinary asymmetric quantum codes are constructed in this paper. These new quantum codes are derived from the Calderbank-Shor-Steane (CSS) construction as well as the Hermitian construction applied respectively to two classical nested Bose-Chaudhuri-Hocquenghem (BCH) codes where one of them are additionally Euclidean (Hermitian) dual-containing. The asymmetric codes constructed here have parameters better than the ones available in the literature.     

Structure of 2D Topological Stabilizer Codes

We provide a detailed study of the general structure of translationally invariant two-dimensional topological stabilizer quantum error correcting codes, including subsystem codes. We show that they can be understood in terms of the homology of string operators that carry a certain topological charge. In subsystem codes, two dual kinds of charges appear. We prove that two non-chiral codes are equivalent under local transformations iff they have isomorphic topological charges. Our approach emphasizes local properties over global ones.     

New Entanglement-Assisted Quantum MDS Codes Derived From Generalized Reed-Solomon Codes

International Journal of Theoretical Physics - Entanglement-assisted quantum error-correcting codes (abbreviate to EAQECCs) expand the usual paradigm of quantum error correction by allowing two...     

Two Families of Entanglement-Assisted Quantum MDS Codes from Cyclic Codes

International Journal of Theoretical Physics - With entanglement-assisted (EA) formalism, arbitrary classical linear codes are allowed to transform into EAQECCs by using pre-shared entanglement...