1-Perfect Uniform and Distance Invariant Partitions

Let F ⁿ be the n-dimensional vector space over ℤ₂. A (binary) 1-perfect partition of F ⁿ is a partition of F ⁿ into (binary) perfect single error-correcting codes or 1-perfect codes. We define two metric properties for 1-perfect partitions: uniformity and distance invariance. Then we prove the equivalence between these properties and algebraic properties of the code (the class containing the zero vector). In this way, we characterize 1-perfect partitions obtained using 1-perfect translation invariant and not translation invariant propelinear codes. The search for examples of 1-perfect uniform but not distance invariant partitions enabled us to deduce a non-Abelian propelinear group structure for any Hamming code of length greater than 7. Received: March 6, 2000; revised version: November 30, 2000     

Matrix-Product Codes over ? q

Codes C ₁ ,…,C _M of length n over ? _q and an M × N matrix A over ? _q define a matrix-product code C = [C ₁ …C _M ] ·A consisting of all matrix products [c ₁ … c _M ] ·A. This generalizes the (u|u+v)-, (u+v+w|2u+v|u)-, (a+x|b+x|a+b+x)-, (u+v|u-v)- etc. constructions. We study matrix-product codes using Linear Algebra. This provides a basis for a unified analysis of |C|, d(C), the minimum Hamming distance of C, and C ^⊥. It also reveals an interesting connection with MDS codes. We determine |C| when A is non-singular. To underbound d(C), we need A to be `non-singular by columns (NSC)'. We investigate NSC matrices. We show that Generalized Reed-Muller codes are iterative NSC matrix-product codes, generalizing the construction of Reed-Muller codes, as are the ternary `Main Sequence codes'. We obtain a simpler proof of the minimum Hamming distance of such families of codes. If A is square and NSC, C ^⊥ can be described using C ₁ ^⊥, …,C _M ^⊥ and a transformation of A. This yields d(C ^⊥). Finally we show that an NSC matrix-product code is a generalized concatenated code. Received: July 20, 1999; revised version: August 27, 2001     

On cyclic self-dual codes

We investigate cyclic self-dual codes over \mathbbF_2^r{\mathbb{F}_{2^{r}}} . We give a decomposition of a repeated-root cyclic codes over \mathbbF_p^r{\mathbb{F}_{p^{r}}} . The decomposition is used to analyze cyclic self-dual codes over \mathbbF_2^r{\mathbb{F}_{2^{r}}} . We obtain a necessary and sufficient condition for the existence of nontrivial cyclic self-dual codes over \mathbbF_2^r{\mathbb{F}_{2^{r}}} , and prove that all cyclic self-dual codes over \mathbbF_2^r{\mathbb{F}_{2^{r}}} are Type I. Finally we classify cyclic self-dual codes of some lengths over \mathbbF₄{\mathbb{F}_{4}} , \mathbbF₈{\mathbb{F}_{8}} , and \mathbbF₁₆{\mathbb{F}_{16}} .     

The a-invariant of some Reed-Muller Codes

The a-invariant, the defining ideal and the dimension of Reed-Muller codes arising from some subsets of the projective n-space over a finite field are determined. In some cases the minimal distance of the code is given. Received: May 20, 1998; revised version: April 27, 1999     

Molecular Dynamics Simulations of Ba1 – x Gd x F2 + x Solid Solutions over a Wide Temperature Range: II. Structural Characteristics and Fluoride Ion Diffusion

The atomic-scale structure (pair correlation functions and anion-density distribution) and fluoride-ion diffusion in Ba_{1 – x} Gd _x F_{2 + x} solid solutions are investigated over a wide temperature range, including the superionic transition, by molecular dynamics simulations with Born–Mayer–Huggins pair potentials. According to the simulation results for x > 0.1, both the 4b(cubic site symmetry) and 48i positions accommodate fluorine interstitials (F^– _i). Below the superionic transition, the migration of F^– ions at low doping levels is dominated by noncollinear jumps between lattice and interstitial sites. Computer simulations confirm that trigonal Gd³⁺–F^– _i dipolar complexes prevail over tetragonal complexes. It is shown using elements of Monte Carlo simulation that the rare-earth ions in the solid solutions have a tendency to aggregate.     

The lower bound on the second-order nonlinearity of a class of Boolean functions with high nonlinearity

The r-th order nonlinearity of Boolean functions is an important cryptographic criterion associated with some attacks on stream and block ciphers. It is also very useful in coding theory, since it is related to the covering radii of Reed-Muller codes. By investigating the lower bound of the nonlinearity of the derivative of the function f, this paper tightens the lower bound of the second-order nonlinearity of a class of Boolean functions over F2n{F_{2^n}} with high nonlinearity in the form f(x) = tr(λ x d ), where l ? F2r*, d=22r+2r+1{\lambda\in F_{2^r}^*, d=2^{2r}+2^{r}+1} and n = 4r.     

Linkage and Codes on Complete Intersections

This note is meant to be an introduction to cohomological methods and their use in the theory of error-correcting codes. In particular we consider evaluation codes on a complete intersection. The dimension of the code is determined by the Koszul complex for X² and a lower bound for the minimal distance is obtained through linkage. By way of example our result fits the well-known formula for the minimal distance of the Generalized Reed-Muller code. Keywords:Liason, Linkage, Complete intersections, Error-correcting codes, Generalized Reed-Muller codesPart of this work was done while visiting Institut de Mathématique de Luminy, 163 avenue de Luminy, Case 907, 13288 Marseille CEDEX 9, France     

Distribution of 3d Activator Ions over C i and C s Isomorphic Positions in Olivine-Structure Crystals

The effect of the olivine structure and properties of 3d activator ions on their distribution over the isomorphic positions C _i and C _s is analyzed. It is shown that the choice of a higher symmetry subgroup among the subgroups of the space group of olivine (Pnma) corresponding to the structures formed by ions in either the C _i or C _s position and Brown's principle lead to a higher probability of incorporation of the activator into position C _i. In terms of the crystal-field potential, this selectivity is due to the deeper potential minimum in position C _i. The angular distribution of electron density in the ground state of 3d ions influences their distribution over positions C _i and C _s in weak crystal fields through the strength of chemical bonding. These conclusions are supported by experimental data for olivine-structure crystals: forsterite and pyroxene activated with iron-group ions and alexandrite.     

The effect of combined diffusion and kinetic transport barriers on multi-phase solid state reactions with a vapour reactant

Analytical models are presented for the rates of layer thickness growth of MoSi₂ and of Mo₅Si₃ that form by reaction of vapour-supplied Si with Mo or with partially silicided Mo. The models are applicable to other systems. Coupling of the diffusive flux of the reactive species Si with the rate of the chemical reactions determines the growth kinetics. The rate of chemical reaction is assumed to be proportional to the magnitude of a discontinuity in the Si activity at the physical boundary where the silicide reaction is occurring. Various combinations of diffusive versus chemical-kinetics-dominated transport in the two phases which grow in tandem are found to affect the functional dependence of the growth kinetics on time. Models include cases in which the host solid is heterogeneous, as occurs when the average composition of the host lies in a poly-phase region of the phase diagram.Nomenclature p porosity, (For examplep ₁ is the porosity of the MoSi₂ layer.) - q 1 — p - v _i volume fraction ofi in the unsilicided compact - V _i theoretical molar volume ofi - _i theoretical volume per gram atom of Si ini - _i (1 —p _i/_i ing atoms of Si per unit volume ini - D _i Diffusion coefficient of Si ini - K _i Kinetic rate constant for conversion of phasei to phasei+ 1 in g.At cm^–2s^–1 per C - J _i Flux of Si through phasei in g.Atcm^–2s^–1 - C ₀ in MoSi₂ at outer surface in equilibrium with Si vapour - C ₁ in MoSi₂ at boundary in equilibrium with Mo₅Si₃ - C ₂] in Mo₅Si₃ at boundary in equilibrium with MoSi₂ - C ₃ in Mo₅Si₃ at boundary in equilibrium with Mo - C ₄ in Mo at boundary in equilibrium with Mo₅Si₃ - C ₁ in MoSi₂ at boundary with Mo₅Si₃, but at a higher Si activity than the equilibrium value - C ₃ in Mo₅Si₃ at boundary with Mo, but at a higher Si activity than equilibrium value - C_ij C _i —C _j - A(Si) Atomic weight of Si - T elapsed time since start of siliciding in seconds - X, Y thicknesses of MoSi₂, Mo₅Si₃ layers in cm     

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     

Annealing and photobleaching effects on the absorption spectra of gamma-irradiated BaFI crystals

We have studied processes induced in gamma-irradiated BaFI crystals by illumination and thermal annealing and identified defect centers responsible for the observed absorption bands. We assign the 205-nm absorption band to α(F⁻) centers, the 230-nm band to α(I⁻) centers, the 270-nm band to F_i ⁰ centers, the 370-nm band to I_i⁰ centers, the bands around 480 nm to F(F⁻) centers, and the 610-nm band to F(I⁻) centers. The proposed assignment accounts well for the radiation-induced processes identified in the crystals.     

Description of loadings and screenings of cracks with the aid of universal weight functions

The stress intensity vector K _i is defined as the limiting behaviour of the stress near the tip of a crack, the stress components being proportional to r ^?1/2 for any external loading. Internal stresses caused by dislocations show the same power dependence at the crack tip; the stress intensity associated with a loading can thus be screened (or amplified) by a plastic zone. Since for any particular specimen and crack geometry the stress intensity vector must be a functional of the loading and screening which are of vectorial character (lines of forces f _i or dislocations with a Burgers vector b _i) one can define two tensorial weight functions, one for screenings, D _si(x, a), and one for forces, F _si(x, a), so that the stress intensity K _s can be found by integration over the product of weight functions and dislocation or force density. In order to find the weight functions the displacement field and the Airy stress vector must be known for some, completely arbitrary, loading or screening.     

Gamma-induced defects in BaFI crystals

We have studied gamma-induced defects in BaFI crystals. To gain insight into the origin of the observed induced absorption bands, we analyzed the dose dependence of the concentration of color centers. The results lead us to assign the 205-nm absorption band to α(F⁻) centers, the 270-nm band to F_i⁰ centers, the 370-nm band to I_i⁰ centers, the bands around 480 nm to F(F⁻) centers, and the 610-nm band to F(I⁻) centers.     

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.     

Generalized quasi-cyclic codes over Galois rings: structural properties and enumeration

For R a Galois ring and m ₁, . . . , m _l positive integers, a generalized quasi-cyclic (GQC) code over R of block lengths (m ₁, m ₂, . . . , m _l ) and length ?_i=1^lm_i{\sum_{i=1}^lm_i} is an R[x]-submodule of R[x]/(x^m₁-1)×?×R[x]/(x^m_l-1){R[x]/(x^{m_1}-1)\times\cdots \times R[x]/(x^{m_l}-1)}. Suppose m ₁, . . . , m _l are all coprime to the characteristic of R and let {g ₁, . . . , g _t } be the set of all monic basic irreducible polynomials in the factorizations of x^m_i-1{x^{m_i}-1} (1 ≤ i ≤ l). Then the GQC codes over R of block lengths (m ₁, m ₂, . . . , m _l ) and length ?_i=1^lm_i{\sum_{i=1}^lm_i} are identified with G₁×?×G_t{{\mathcal G}_1\times\cdots\times {\mathcal G}_t}, where G_j{{\mathcal G}_j} is an R[x]/(g _j )-submodule of (R[x]/(g_j))^n_j{(R[x]/(g_j))^{n_j}}, where n _j is the number of i for which g _j appears in the factorization of x^m_i-1{x^{m_i}-1} into monic basic irreducible polynomials. This identification then leads to an enumeration of such GQC codes. An analogous result is also obtained for the 1-generator GQC codes. A notion of a parity-check polynomial is given when R is a finite field, and the number of GQC codes with a given parity-check polynomial is determined. Finally, an algorithm is given to compute the number of GQC codes of given block lengths.     

Addition of the Lewis Acid Zn(C6F5)2 Enables Organic Transistors with a Maximum Hole Mobility in Excess of 20 cm2 V−1 s−1

Incorporating the molecular organic Lewis acid tris(pentafluorophenyl)borane [B(C₆F₅)₃] into organic semiconductors has shown remarkable promise in recent years for controlling the operating characteristics and performance of various opto/electronic devices, including, light‐emitting diodes, solar cells, and organic thin‐film transistors (OTFTs). Despite the demonstrated potential, however, to date most of the work has been limited to B(C₆F₅)₃ with the latter serving as the prototypical air‐stable molecular Lewis acid system. Herein, the use of bis(pentafluorophenyl)zinc [Zn(C₆F₅)₂] is reported as an alternative Lewis acid additive in high‐hole‐mobility OTFTs based on small‐molecule:polymer blends comprising 2,7‐dioctyl[1]benzothieno [3,2‐b][1]benzothiophene and indacenodithiophene–benzothiadiazole. Systematic analysis of the materials and device characteristics supports the hypothesis that Zn(C₆F₅)₂ acts simultaneously as a p‐dopant and a microstructure modifier. It is proposed that it is the combination of these synergistic effects that leads to OTFTs with a maximum hole mobility value of 21.5 cm² V^?1 s^?1. The work not only highlights Zn(C₆F₅)₂ as a promising new additive for next‐generation optoelectronic devices, but also opens up new avenues in the search for high‐mobility organic semiconductors.     

Estimating the Degrees of Freedom for Linear Combinations of Mean Squares by Satterthwaite's Formula

In calculating the variance of a mean or in constructing an approximate F-test, it is frequently necessary, particularly with unbalanced data, to form a linear function of mean squares, MS = Σa _i, where the a_i are known constants. Satterthwaite [1] suggests that MS is approximately distributed as x ² _f E(MS)/f_t where the degrees of freedom is estimated by, f = (MS)²/Σ[(a_iMS_i )²/f_i ], where f_i is the degrees of freedom associated with MS_i . Satterthwaite remarks that caution should be used in applying this formula where some of the a_i are negative. The primary purpose of this paper is to investigate the validity of Satterthwaite's procedure when some MS_i are subtracted. Due to the mathematical complexity of the distribution of MS, the problem is studied by a computer simulation.     

Reed-Muller Codes on Complete Intersections

By using results and techniques from commutative algebra such as the vanishing ideal of a set of points, its a-invariant, the Hilbert polynomial and series, as well as finite free resolutions and the canonical module, some results about Reed-Muller codes defined on a zero-dimensional complete intersection in the n-projective dimensional space are given. Several examples of this class of codes are presented in order to illustrate the ideas. Received: March 11, 1999; revised version: November 6, 2000     

Trellis complexity and pseudoredundancy of relative two-weight codes

Relative two-weight codes have been studied due to their applications to wiretap channel and secret sharing. It has been shown that these codes form a large family, which includes dual Hamming codes and subcodes of punctured Reed-Muller codes as special instances. This work studies the properties of relative two-weight codes with regard to efficient decoding. More specifically, the trellis complexity, which determines the complexity of Viterbi algorithm based decoding and pseudoredundancy that measures the performance and complexity of linear programming decoding are studied for relative two-weight codes. Separating properties of these codes have been identified and proved first. Based on the results of separating properties, the trellis complexity of binary relative two-weight codes is fully determined. An upper bound on the pseudoredundancy of binary relative two-weight codes is derived.     

Generalised critical free-surface flows

Nonlinear waves in a forced channel flow are considered. The forcing is due to a bottom obstruction. The study is restricted to steady flows. A weakly nonlinear analysis shows that for a given obstruction, there are two important values of the Froude number, which is the ratio of the upstream uniform velocity to the critical speed of shallow water waves, F _C>1 and F _L<1 such that: (i) when F<F _L, there is a unique downstream cnoidal wave matched with the upstream (subcritical) uniform flow; (ii) when F=F _L, the period of the cnoidal wave extends to infinity and the solution becomes a hydraulic fall (conjugate flow solution) – the flow is subcritical upstream and supercritical downstream; (iii) when F>F _C, there are two symmetric solitary waves sustained over the site of forcing, and at F=F _C the two solitary waves merge into one; (iv) when F>F _C, there is also a one-parameter family of solutions matching the upstream (supercritical) uniform flow with a cnoidal wave downstream; (v) for a particular value of F>F _C, the downstream wave can be eliminated and the solution becomes a reversed hydraulic fall (it is the same as solution (ii), except that the flow is reversed!). Flows of type (iv), including the hydraulic fall (v) as a special case, are computed here using the full Euler equations. The problem is solved numerically by a boundary-integral-equation method due to Forbes and Schwartz. It is confirmed that there is a three-parameter family of solutions with a train of waves downstream. The three parameters can be chosen as the Froude number, the obstruction size and the wavelength of the downstream waves. This three-parameter family differs from the classical two-parameter family of subcritical flows (i) but includes as a particular case the hydraulic falls (ii) or equivalently (v) computed by Forbes.