Large Minimum Distance Research Articles

Application of Hermitian self-orthogonal GRS codes to some quantum MDS codes

The construction of quantum maximum distance separable (abbreviated to MDS) error-correcting codes has become one of the major concerns in quantum coding theory. In this paper, we further generalize the approach developed in the previous paper, and construct several new classes of Hermitian self-orthogonal generalized Reed-Solomon (GRS) codes. By employing these classical MDS codes, we obtain several classes of quantum MDS codes with large minimum distance. It turns out that our quantum MDS codes exhibited here have less constraints on the selection of code length and some of them have not been constructed before, and in some cases, have larger minimum distance than previous literature. Meanwhile, about half of the distance parameters of our codes are greater than q2+1.

New entanglement-assisted quantum MDS codes

Entanglement-assisted quantum error-correcting codes (EAQECCs) can be obtained from arbitrary classical linear codes based on the entanglement-assisted stabilizer formalism, which greatly promoted the development of quantum coding theory. In this paper, we construct several families of [Formula: see text]-ary entanglement-assisted quantum maximum-distance-separable (EAQMDS) codes of lengths [Formula: see text] with flexible parameters as to the minimum distance [Formula: see text] and the number [Formula: see text] of maximally entangled states. Most of the obtained EAQMDS codes have larger minimum distances than the codes available in the literature.

A mΘ spectrum of Reed-Muller codes

In this paper, we construct mΘ spectrum of Reed Muller codes, a family of mΘ linear codes [5] with large mΘ minimum distance. First, we cover the necessary material on the mΘ Boolean algebra. We then define the mΘ spectrum of Reed Muller codes RM Θ (r, m) as certain subspaces of the mΘ Boolean algebra . We determinate the parameters of RM Θ (r, m).

SCMA Codebook Design Based on Uniquely Decomposable Constellation Groups

Sparse code multiple access (SCMA), which helps improve spectrum efficiency (SE) and enhance connectivity, has been proposed as a non-orthogonal multiple access (NOMA) scheme for 5G systems. In SCMA, codebook design determines system overload ratio and detection performance at a receiver. In this paper, an SCMA codebook design approach is proposed based on uniquely decomposable constellation group (UDCG). We show that there are N+1 ( N ≥ 1) constellations in the proposed UDCG, each of which has M (M ≥ 2) constellation points. These constellations are allocated to users sharing the same resource. Combining the constellations allocated on multiple resources of each user, we can obtain UDCG-based codebook sets. Bit error ratio (BER) performance will be discussed in terms of coding gain maximization with superimposed constellations and UDCG-based codebooks. Simulation results demonstrate that the superimposed constellation of each resource has large minimum Euclidean distance (MED) and meets uniquely decodable constraint. Thus, BER performance of the proposed codebook design approach outperforms that of the existing codebook design schemes in both uncoded and coded SCMA systems, especially for large-size codebooks.

Some Classes of New Quantum MDS and Synchronizable Codes Constructed From Repeated-Root Cyclic Codes of Length 6ps

In this paper, we use the CSS and Steane’s constructions to establish quantum error-correcting codes (briefly, QEC codes) from cyclic codes of length $6p^{s}$ over $\mathbb F_{p^{m}}$ . We obtain several new classes of QEC codes in the sense that their parameters are different from all the previous constructions. Among them, we identify all quantum MDS (briefly, qMDS) codes, i.e., optimal quantum codes with respect to the quantum Singleton bound. In addition, we construct quantum synchronizable codes (briefly, QSCs) from cyclic codes of length $6p^{s}$ over $\mathbb F_{p^{m}}$ . Furthermore, we give many new QSCs to enrich the variety of available QSCs. A lot of them are QSCs codes with shorter lengths and much larger minimum distances than known non-primitive narrow-sense BCH codes.

A Novel Niederreiter-like cryptosystem based on the (u|u+υ)-construction codes

In this paper, we present a new variant of the Niederreiter Public Key Encryption (PKE) scheme which is resistant against recent attacks. The security is based on the hardness of the Rank Syndrome Decoding (RSD) problem and it presents a (u|u+υ)-construction code using two different types of codes: Ideal Low Rank Parity Check (ILRPC) codes andλ-Gabidulin codes. The proposed encryption scheme benefits are a larger minimum distance, a new efficient decoding algorithm and a smaller ciphertext and public key size compared to the Loidreau’s variants and to its IND-CCA secure version.

On the Construction of Multitype Quasi-Cyclic Low-Density Parity-Check Codes With Different Girth and Length

Multitype quasi-cyclic (QC) low-density parity-check (LDPC) codes are a class of protograph LDPC codes lifted cyclically from protographs with multiple edges, represented by two weight and slope matrices. For a given weight-matrix, an approach is proposed to find the maximum-achievable girth $g_{\max }$ of the corresponding multitype QC-LDPC codes by some inevitable chains having less complexity than the existing methods. This advantage leads to some new patterns of the weight matrices such that the corresponding codes have some improvements in terms of the maximum-achievable girths or the minimum-distance upper-bounds. In continue, for a given weight-matrix with maximum-achievable girth $g_{\max }$ , some slope-matrices are constructed by a depth-first search algorithm for which the corresponding multitype QC-LDPC codes with even girth $g$ , $g\le g_{\max }$ , have smaller lengths, higher rates, or larger minimum-distances than the state-of-the-art achievements. Simulation results show that the constructed codes have some error-rate advantages than PEG, random-like, CCSDS, and 802.11n/ac IEEE standard LDPC codes.

Mitigating the Impact of Noise on Nonlinear Frequency Division Multiplexing

In the past years, nonlinear frequency division multiplexing (NFDM) has been investigated as a potentially revolutionary technique for nonlinear optical fiber communication. However, while NFDM is able to exploit the Kerr nonlinearity, its performance lags behind that of conventional systems. In this work, we first highlight that current implementations of NFDM are strongly suboptimal, and, consequently, oversensitive to noise: the modulation does not ensure a large minimum distance between waveforms, while the detection is not tailored to the statistics of noise. Next, we discuss improved detections strategies and modulation techniques, proposing some effective approaches able to improve NFDM. Different flavors of NFDM are compared through simulations, showing that (i) the NFDM performance can be significantly improved by employing more effective detection strategies, with a 5.6 dB gain in Q-factor obtained with the best strategy compared to the standard strategy; (ii) an additional gain of 2.7 dB is obtained by means of a simple power-tilt modulation strategy, bringing the total gain with respect to standard NFDM to 8.3 dB; and (iii) under some parameters range (rate efficiency η≤30%), the combination of improved modulation and detection allows NFDM to outperform conventional systems using electronic dispersion compensation.

Superposed constellation design for spatial multiplexing visible light communication systems.

Superposed constellation for spatial multiplexing visible light communication (VLC) systems has recently attracted significant attention as a promising method to alleviate the effect of nonlinearity on light-emitting diodes (LEDs) that has achieved significant performance improvements in VLC systems. A large minimum Euclidean distance (ED) value of the superposed signals can be achieved at the receiver side by employing optimal power ratios transmitted from different LEDs. However, the power allocation strategy using fixed shaped sub-constellations at the LEDs has the inherent limitation of low optimization dimensions. In this paper, we propose optimizing the sub-constellations at individual LEDs instead of just the power allocation coefficients to further increase the minimum ED value. Moreover, intra-ED and inter-ED terms are used to reduce the complexity of the optimization solving process. The simulation results show that the proposed constellation design scheme can improve the symbol error rate performance compared to the conventional one, and the right choice of pre-defined shape of the sub-constellations at the LEDs has an important role in the optimization process.

Construction of entanglement-assisted quantum MDS codes

Entanglement-assisted quantum error correction codes (EAQEECs) can improve the reliability of transmission, which make use of pre-existing entanglement between two parties. In this paper, we construct two classes of entanglement-assisted quantum maximum distance separable (EAQMDS) codes via generalized Reed-Solomon (GRS) codes. Consequently, the results show that some of these EAQMDS codes have much larger minimum distance than ones available in the literature. Meanwhile, many of these EAQMDS codes are new in the sense that the parameters of these codes are not covered by the previously known ones. ()

Optimal Entanglement-Assisted Quantum Codes With Larger Minimum Distance

Entanglement-assisted quantum error-correcting code (EAQECC) is a new-type quantum code and can be constructed from arbitrary linear codes by sharing entangled states c in advance. In this letter, we explore a family of entanglement-assisted quantum error-correcting codes (EAQECCs) with length (q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> - 1)/12 from negacyclic codes. Compared with known results in the literatures, our EAQECCs are new and have larger minimum distance.

New classes of entanglement-assisted quantum MDS codes

In this paper, two new classes of entanglement-assisted quantum MDS codes (EAQMDS codes for short) with length n being a factor of $$q^2\pm 1$$ are presented via cyclic codes over finite fields of odd characteristic. Among our constructions, there are several EAQMDS codes with new parameters which have never been reported. Moreover, some of them have much larger minimum distance than known results.

Non-Split Toric BCH Codes on Singular del Pezzo Surfaces

In the article we construct low-rate non-split toric $q$ -ary codes on some singular surfaces. More precisely, we consider non-split toric cubic and quartic del Pezzo surfaces, whose singular points are $\mathbb {F}_{\!q}$ -conjugate. Our codes turn out to be BCH ones with sufficiently large minimum distance $d$ . Indeed, we prove that $d - d^{*} \geqslant q - \lfloor 2\sqrt {q} \rfloor - 1$ , where $d^{*}$ is the designed minimum distance. In other words, we significantly improve upon BCH bound. On the other hand, the defect of the Griesmer bound for the new codes is $\leqslant \lfloor 2\sqrt {q} \rfloor - 1$ , which also seems to be quite good. It is worth noting that to better estimate $d$ we actively use the theory of elliptic curves over finite fields.

On maximin distance and nearly orthogonal Latin hypercube designs

Maximin distance Latin hypercube designs (LHDs) are frequently used in computer experiments, but their constructions are challenging. In this paper, we present some new results connecting maximin L2-distance optimality and near orthogonality for mirror-symmetric LHDs. We further propose a simple and effective method for constructing nearly orthogonal LHDs that can yield almost the largest minimum distance. The obtained designs with small and medium sizes are tabulated and their superior performances are illustrated via comparisons.

New entanglement-assisted quantum MDS codes with length $$n=\frac{q^2+1}{5}$$

The entanglement-assisted stabilizer formalism can transform arbitrary classical linear codes into entanglement-assisted quantum error correcting codes (EAQECCs). In this work, we construct some new entanglement-assisted quantum MDS (EAQMDS) codes with length $n=\frac{q^2+1}5$ from cyclic codes. Compared with all the previously known parameters with the same length, all of them have flexible parameters and larger minimum distance.

New entanglement-assisted quantum MDS codes with larger minimum distance

In this paper, we construct some new entanglement-assisted quantum maximum distance separable (EAQMDS) codes with lengths $$n=q^2+1$$ and $$n=(q^2+1)/2$$ from negacyclic MDS codes and constacyclic MDS codes, respectively. All of them have flexible parameters. These EAQMDS codes we constructed have larger minimum distance and contain the known EAQMDS codes with same length in previous papers.

Some construction of entanglement-assisted quantum MDS codes

Entanglement-assisted quantum error-correcting codes expand the usual paradigm of quantum error correction by allowing two parties to make use of pre-shared entanglement. This entanglement can increase either the rate of communication or the number of correctable errors. By employing generalized Reed–Solomon codes, we construct several classes of entanglement-assisted quantum maximum distance separable (EAQMDS) codes in this paper. Consequently, the results show that many of these EAQMDS codes have much larger minimum distance than ones available in the literature. Meanwhile, some of these EAQMDS codes are new in the sense that the parameters of these codes are not covered by the previously known ones, whose required number of maximally entangled states is more flexible.

A novel construction for quantum stabilizer codes based on binary formalism

In this research, we propose a novel construction of quantum stabilizer code based on a binary formalism. First, from any binary vector of even length, we generate the parity-check matrix of the quantum code from a set composed of elements from this vector and its relations by shifts via subtraction and addition. We prove that the proposed matrices satisfy the condition constraint for the construction of quantum codes. Finally, we consider some constraint vectors which give us quantum stabilizer codes with various dimensions and a large minimum distance with code length from six to twelve digits.

Minimum distances for wind turbines: A robustness analysis of policies for a sustainable wind power deployment

The deployment of wind power is a major contribution to the decarbonisation of societies. Yet, wind turbines can cause some negative externalities to humans and nature. These largely depend on the spatial allocation of the wind turbines. Therefore the question is how to design policies that minimise the social costs of wind power generation which are defined as the sum of production and external costs. An instrument which is used in Germany and elsewhere to control the externalities of wind turbines is the prescription of minimum distances to sensitive landscape features like human settlements and bird nests. The efficient (i.e. minimising social costs) magnitude of such minimum distances, however, depends on uncertain parameters. We apply a robustness analysis to an ecological-economic model for the assessment of the social costs of wind power deployment in order to identify policies, each defined by certain minimum distances, which are favourable within wide ranges of various uncertain parameters. In the examined study region in Germany, rather large minimum distances to nests of the red kite (a raptor bird) and moderate minimum distances to settlements turn out to be most favourable taken the considered uncertainties into account.

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

Entanglement-assisted quantum error correcting codes (EAQECCs) can be derived from arbitrary classical linear codes. However, it is a very difficult task to determine the number of entangled states required. In this work, using the method of the decomposition of the defining set of constacyclic codes, we construct two families of q-ary entanglement-assisted quantum MDS (EAQMDS) codes based on classical constacyclic MDS codes by exploiting less pre-shared maximally entangled states. We show that a class of q-ary EAQMDS have minimum distance upper bound greater than q. Some of them have much larger minimum distance than the known quantum MDS (QMDS) codes of the same length. Most of these q-ary EAQMDS codes are new in the sense that their parameters are not covered by the codes available in the literature.