Some new binary quasi-cyclic codes from codes over the ring \mathbb F _2+u\mathbb F _2+v\mathbb F _2+uv\mathbb F _2

We consider quasi-cyclic codes over the ring $\mathbb{F }_2+u\mathbb{F }_2+v\mathbb{F }_2+uv\mathbb{F }_2$ , a finite non-chain ring that has been recently studied in coding theory. The Gray images of these codes are shown to be binary quasi-cyclic codes. Using this method we have obtained seventeen new binary quasi-cyclic codes that are new additions to the database of binary quasi-cyclic codes. Moreover, we also obtain a number of binary quasi-cyclic codes with the same parameters as best known binary linear codes that otherwise have more complicated constructions.     

Cyclic codes over $${\mathbb {F}}_{2^m}[u]/\langle u^k\rangle $$ of oddly even length

Let \({\mathbb {F}}_{2^m}\) be a finite field of characteristic 2 and \(R={\mathbb {F}}_{2^m}[u]/\langle u^k\rangle ={\mathbb {F}}_{2^m} +u{\mathbb {F}}_{2^m}+\ldots +u^{k-1}{\mathbb {F}}_{2^m}\) (\(u^k=0\)) where \(k\in {\mathbb {Z}}^{+}\) satisfies \(k\ge 2\). For any odd positive integer n, it is known that cyclic codes over R of length 2n are identified with ideals of the ring \(R[x]/\langle x^{2n}-1\rangle \). In this paper, an explicit representation for each cyclic code over R of length 2n is provided and a formula to count the number of codewords in each code is given. Then a formula to calculate the number of cyclic codes over R of length 2n is obtained. Moreover, the dual code of each cyclic code and self-dual cyclic codes over R of length 2n are investigated.     

One weight $$\mathbb {Z}_2\mathbb {Z}_4$$ additive codes

We study one weight \(\mathbb {Z}_2\mathbb {Z}_4\) additive codes. It is shown that the image of an equidistant \(\mathbb {Z}_2\mathbb {Z}_4\) code is a binary equidistant code and that the image of a one weight \(\mathbb {Z}_2\mathbb {Z}_4\) additive code, with nontrivial binary part, is a linear binary one weight code. The structure and possible weights for all one weight \(\mathbb {Z}_2\mathbb {Z}_4\) additive codes are described. Additionally, a lower bound for the minimum distance of dual codes of one weight additive codes is obtained.     

Bounds on the minimum distance of the duals of extended BCH codes over\mathbb{F}_p

LetC be an extended cyclic code of lengthp ^m over . The border ofC is the set of minimal elements (according to a partial order on [0,p ^m –1]) of the complement of the defining-set ofC. We show that an affine-invariant code whose border consists of only one cyclotomic coset is the dual of an extended BCH code if, and only if, this border is the cyclotomic coset, sayF(t, i), ofp ^t –1–i, with 1 t m and 0 i < p–1. We then study such privileged codes. We first make precize which duals of extendedBCH codes they are. Next, we show that Weil's bound in this context gives an explicit formula; that is, the couple (t, i) fully determines the value of the Weil bound for the code with borderF(t, i). In the case where this value is negative, we use the Roos method to bound the minimum distance, greatly improving the BCH bound.     

On the arithmetic of the endomorphisms ring {{\rm End}(\mathbb{Z}_{p} \times \mathbb{Z}_{p^{2}})}

For a prime number p, Bergman (Israel J Math 18:257–277, 1974) established that End(\mathbbZ_p×\mathbbZ_p²){{\rm End}(\mathbb{Z}_{p}\times \mathbb{Z}_{p^{2}})} is a semilocal ring with p ⁵ elements that cannot be embedded in matrices over any commutative ring. We identify the elements of End(\mathbbZ_p ×\mathbbZ_p²){{\rm End}(\mathbb{Z}_{p} \times \mathbb{Z}_{p^{2}})} with elements in a new set, denoted by E _p , of matrices of size 2 × 2, whose elements in the first row belong to \mathbbZ_p{\mathbb{Z}_{p}} and the elements in the second row belong to \mathbbZ_p²{\mathbb{Z}_{p^{2}}}; also, using the arithmetic in \mathbbZ_p{\mathbb{Z}_{p}} and \mathbbZ_p²{\mathbb{Z}_{p^{2}}}, we introduce the arithmetic in that ring and prove that the ring End(\mathbbZ_p ×\mathbbZ_p²){{\rm End}(\mathbb{Z}_{p} \times \mathbb{Z}_{p^{2}})} is isomorphic to the ring E _p . Finally, we present a Diffie-Hellman key interchange protocol using some polynomial functions over E _p defined by polynomials in \mathbbZ[X]{\mathbb{Z}[X]}.     

Cryptanalysis of a key exchange protocol based on the endomorphisms ring End{(\mathbb{Z}_{p} \times \mathbb{Z}_{p^2})}

Climent et?al. (Appl Algebra Eng Commun Comput 22:91?C108, 2011) identified the elements of the endomorphisms ring End ${(\mathbb{Z}_p \times \mathbb{Z}_{p^2})}$ with elements in a set, E _p , of matrices of size 2?× 2, whose elements in the first row belong to ${\mathbb{Z}_{p}}$ and the elements in the second row belong to ${\mathbb{Z}_{p^2}}$ . By taking advantage of matrix arithmetic, they proposed a key exchange protocol using polynomial functions over E _p defined by polynomials in ${\mathbb{Z}[X]}$ . In this note, we show that this protocol is insecure; it can be broken by solving a set of 10 consistent homogeneous linear equations in 8 unknowns over ${\mathbb{Z}_{p^2}}$ .     

Duadic Codes over F2 + uF2

Duadic codes over F ₂ + u F ₂ are introduced as abelian codes by their zeros. This is the function field analogue of duadic codes over Z ₄ introduced recently by Langevin and Solé. They produce binary self-dual codes via a suitable Gray map. Their binary images are themselves abelian, thus generalizing a result of van Lint for cyclic binary codes of even length. We classify them in modest lengths and exhibit interesting non-cyclic examples. Received: April 26, 2000; revised version: May 5, 2001     

Two new permutation polynomials with the form {\left(x^{2^k}+x+\delta\right)^{s}+x} over {\mathbb{F}_{2^n}}

This note presents two new permutation polynomials with the form ${p(x)=\left(x^{2^k}+x+\delta\right)^{s}+x}$ over the finite field ${\mathbb{F}_{2^n}}$ as a supplement of the recent work of Yuan, Ding, Wang and Pieprzyk.     

The trace spectra of polynomial bases for $${\mathbb{F}}_{2^{n}}$$

In this paper we study the trace spectra of polynomial bases for over . Shparlinski showed that there exists a polynomial basis having O(log n) elements of trace one. Here we show that for every t ≤ n, there exists a polynomial basis having t + O(log n) elements of trace one. We also study consequences of our results to the existence of irreducible polynomials of certain weights.     

Evolution of the electrical resistance of \bf{\hbox {YBa}_2\hbox {Cu}_3\hbox {O}_{7-\varvec\delta }} single crystals in the course of long-term aging

Investigated are the changes in the basal-plane electrical resistivity of an optimally doped \(\hbox {YBa}_2\hbox {Cu}_3\hbox {O}_{7-\delta }\) single crystal in the course of long-term aging (17 years) at room temperature in air. In consequence of aging the sample has decomposed into three phases with different temperatures of the superconducting transition, while the transition widths of these phases have increased significantly. The temperature dependence of the electrical resistivity has retained a metallic character. The fluctuation conductivity near the critical temperature is described well by the 3D Aslamazov–Larkin model. In the course of aging significant changes in the scattering characteristics have been observed, whereas the Debye temperature has changed slightly and the transverse coherence length has remained constant.     

High-pressure studies of superconductivity in $$\hbox {BiO}_{0.75}\hbox {F}_{0.25}\hbox {BiS}_{2}$$

\(\hbox {BiO}_{0.75}\hbox {F}_{0.25}\hbox {BiS}_{2}\) crystallizes in tetragonal CeOBiS\(_{2}\) structure (S. G. P4/nmm). We have investigated the effect of pressure on magnetization measurements. Our studies suggest improved superconducting properties in polycrystalline samples of \(\hbox {BiO}_{0.75}\hbox {F}_{0.25}\hbox {BiS}_{2}\). The \(T_{\mathrm{c}}\) in our sample is 5.3 K, at ambient pressure, which is marginal but definite enhancement over \(T_{\mathrm{c}}\) reported earlier (= 5.1 K). The upper critical field \(H_{\mathrm{c}2}\)(0) is greater than 3 T, which is higher than earlier report on this material. As determined from the M–H curve, both \(H_{\mathrm{c}2}\) and \(H_{\mathrm{c}1}\) decrease under external pressure P (0 \(\le P \le \) 1 GPa). We observe a decrease in critical current density and transition temperature on applying pressure in \(\hbox {BiO}_{0.75}\hbox {F}_{0.25}\hbox {BiS}_{2}\).     

Correlations for the Dielectric Constants of $$\hbox {H}_{2}\hbox {S}$$, $$\hbox {SO}_{2}$$SO2, and $$\hbox {SF}_{6}$$SF6

A new method is developed for correlating the static dielectric constant of polar fluids over wide ranges of conditions where few experimental data exist. Molecular dynamics simulations are used to establish the temperature and density dependence of the Kirkwood g-factor, and also the functional form for the increase of the effective dipole moment with density. Most parameters in the model are obtained entirely from simulation; a single proportionality constant is adjusted to obtain agreement with the limited experimental data. The method is applied to hydrogen sulfide (\(\hbox {H}_{2}\hbox {S}\)) and sulfur dioxide \((\hbox {SO}_{2})\), both of which are important in geochemistry but have only a few dielectric data available. The resulting correlations agree well with the available liquid data, obey physical boundary conditions at low density and at high temperature, and interpolate in density and temperature in a physically reasonable manner. In addition, we present a more conventional correlation for the dielectric constant of sulfur hexafluoride, \(\hbox {SF}_{6}\), where more data are available.     

Evaluation of Thermodynamic Models for Predicting Phase Equilibria of $$\hbox {CO}_{2}$$ + Impurity Binary Mixture

For the design and operation of \(\hbox {CO}_{2}\) capture and storage (CCS) processes, equation of state (EoS) models are used for phase equilibrium calculations. Reliability of an EoS model plays a crucial role, and many variations of EoS models have been reported and continue to be published. The prediction of phase equilibria for \(\hbox {CO}_{2}\) mixtures containing \(\hbox {SO}_{2}\), \(\hbox {N}_{2}\), NO, \(\hbox {H}_{2}\), \(\hbox {O}_{2}\), \(\hbox {CH}_{4}\), \(\hbox {H}_{2}\mathrm{S}\), Ar, and \(\hbox {H}_{2}\mathrm{O}\) is important for \(\hbox {CO}_{2}\) transportation because the captured gas normally contains small amounts of impurities even though it is purified in advance. For the design of pipelines in deep sea or arctic conditions, flow assurance and safety are considered priority issues, and highly reliable calculations are required. In this work, predictive Soave–Redlich–Kwong, cubic plus association, Groupe Européen de Recherches Gazières (GERG-2008), perturbed-chain statistical associating fluid theory, and non-random lattice fluids hydrogen bond EoS models were compared regarding performance in calculating phase equilibria of \(\hbox {CO}_{2}\)-impurity binary mixtures and with the collected literature data. No single EoS could cover the entire range of systems considered in this study. Weaknesses and strong points of each EoS model were analyzed, and recommendations are given as guidelines for safe design and operation of CCS processes.     

Structural,elastic, optoelectronic and magnetic properties of $$\mathbf{CdHo }_\mathbf{2}{} \mathbf{S}_\mathbf{4}$$ spinel: a first-principle study

We report the results of the full-potential linearized augmented plane wave (FP-LAPW) calculations on the structural, elastic, optoelectronic and magnetic properties of \(\hbox {CdHo}_{2}\hbox {S}_{4}\) spinel. Both the generalized gradient approximation (GGA) and Trans-Blaha modified Becke-Johnson potential (TB-mBJ) are used to model the exchange-correlation effects. The computed lattice parameter, internal coordinate and bulk modulus are in good agreement with the existing experimental data. According to the calculated elastic moduli, \(\hbox {CdHo}_{2}\hbox {S}_{4}\) is mechanically stable with a ductile nature and a noticeable elastic anisotropy. The ferromagnetic phase of \(\hbox {CdHo}_{2}\hbox {S}_{4}\) is energetically favourable compared to non-magnetic one, with a high magnetic moment of about 8.15 \(\upmu _{\mathrm{B}}\). The calculated band structure demonstrates that the title compound is a direct band gap semiconductor. The TB-mBJ yields a band gap of \(\sim \)1.86 and \(\sim \)2.17 eV for the minority and majority spins, respectively. The calculated optical spectra reveal a strong response in the energy range between the visible light and the extreme UV regions.     

Structures and electronic properties of $$\hbox {W}_{m}\hbox {Cu}_{n}\,(n + m \le 7)$$ clusters

Geometric and electronic structures of \(\hbox {W}_{m}\hbox {Cu}_{n}\, (m + n \le 7)\) cluster have been systematically calculated by density functional theory at the generalized gradient approximation level for ground-state structures. \(\hbox {W}_{m}\hbox {Cu}_{n}\) clusters with \(n = 1, 3, 5\) tend to form bipyramid structures, whereas \(\hbox {WCu}_{n}\) favour planar shapes except for \(\hbox {WCu}_{5}\). The configurations of \(\hbox {W}_{m}\hbox {Cu}_{n}\) clusters are more sensitive to the Cu atoms than the W atoms, while the average atomic binding energies and the total magnetic moments are determined by W atoms. The calculated second-order differences in energies and HOMO–LOMO energy gaps show pronounced odd–even oscillating behaviours. From the Mulliken electron population analysis, we found that Cu 4p and W 6p orbitals exhibit electronic charges and both Cu 4s and W 6s orbitals transfer electronic charges to the W 5d orbital, which lead to the extension of W–Cu bond lengths.     

Investigation on the impact and compression behavior of wet \upgamma -\hbox {Al}_{2}\mathrm{O}_{3} granules

The material behavior of dominant elastic–plastic \(\upgamma \) - \(\hbox {Al}_{2}\mathrm{O}_{3}\) granules has been experimentally studied by means of quasi static compression tests and dynamic impact tests until fracture. The obtained distributions of breakage velocity and specific breakage energy are compared. Thus, velocity dependent influences at stressing like viscous behavior can be derived. Additionally, the influences of particle size and moisture content on the material behavior are investigated.