Bounds on Minimum Distance for Linear Codes over GF(5)

Let [n, k, d; q]-codes be linear codes of length n, dimension k and minimum Hamming distance d over GF(q). Let d ₅(n, k) be the maximum possible minimum Hamming distance of a linear [n, k, d; 5]-code for given values of n and k. In this paper, forty four new linear codes over GF(5) are constructed and a table of d ₅(n, k) k≤ 8, n≤ 100 is presented.     

The number of MDS [7, 3] codes on finite fields of characteristic 2

A code of lengthn, dimensionk and minimum distanced ismaximum distance separable (MDS) ifk+d=n+1. We give the number of MDS codes of length 7 and dimension 3 on finite fields withq elements whereq=2 ^m . In order to get this number, we compute the number of configurations of seven points in the projective plane overF _q , no three of which are collinear.     

Toric Codes over Finite Fields

In this note, a class of error-correcting codes is associated to a toric variety defined over a finite field _q, analogous to the class of AG codes associated to a curve. For small q, many of these codes have parameters beating the Gilbert-Varshamov bound. In fact, using toric codes, we construct a (n,k,d)=(49,11,28) code over ₈, which is better than any other known code listed in Brouwers tables for that n, k and q. We give upper and lower bounds on the minimum distance. We conclude with a discussion of some decoding methods. Many examples are given throughout.     

On cyclic codes which areq-ary images of linear codes

This paper presents a complete characterization of cyclic codes over GF(q) which areq-ary images of linear codes over GF(q ²). New cyclic codes over GF(2 ^r ) are constructed as images of other cyclic codes over GF(2^3r ), for some positive integersr. An application to decoding is given.     

Theory of Plane Copper and Oxygen Nuclear Spin–Lattice Relaxation Rates in Lightly Doped La2− x Sr x CuO4

Using a Mori-Zwanzig projection operator procedure the relaxation function theory of doped two-dimensional Heisenberg antiferromagnetic (AF) system in the paramagnetic state is presented taking into account the hole subsystem as well as both the electron and AF correlations. At low temperatures the main contribution to the nuclear spin–lattice relaxation rate, ⁶³(1/T ₁), of plane ⁶³Cu, arises from the AF fluctuations, and ¹⁷(1/T ₁), of plane ¹⁷O, has the contributions from the wave vectors in the vicinity of (π,π) and small q ∼ 0. The effects of thermal spin-wave damping Γ _q on ¹⁷(1/T ₁) in lightly doped regime are investigated, suggesting either a polynomial of up to third order (not simply (T/J)³) or exponential temperature dependence of Γ _q at low temperatures. It is shown that the theory is able to explain the main features of experimental data on temperature and doping dependence of ^17,63(1/T ₁) in the paramagnetic state of La_{2− x} Sr _x CuO₄ compounds.     

Linear codes with balanced weight distribution

In this paper, we study particular linear codes defined overF _q , with an astonishing property, their weight distribution is balanced, i.e. there is the same number of codewords for each nonzero weight of the code. We call these codesBWD-codes. We first study BWD-codes by means of the Pless identities and we completely characterize the two-weight projective case. We study the class of codes defined under subgroups of the multiplicative group ofF _q ^s , using the Gauss sums. Then, given a primep and an integerN dividingp – 1, we construct all theN-weight BWD-codes of that class. We conclude this paper by some tables of BWD-codes and an open problem.     

Improved Bounds for Quaternary Linear Codes of Dimension 6

In this paper, twenty new codes of dimension 6 are presented which give improved bounds on the maximum possible minimum distance of quaternary linear codes. These codes belong to the class of quasi-twisted (QT) codes, and have been constructed using a stochastic optimization algorithm, tabu search. A table of upper and lower bounds for d ⁴(n,6) is presented for n≤ 200. Received: 20 December 1996 / Accepted: 13 May 1997     

Two “Dual” Families of Nearly-Linear Codes over ℤ p , p odd

Since the paper by Hammons e.a. [1], various authors have shown an enormous interest in linear codes over the ring ℤ₄. A special weight function on ℤ₄ was introduced and by means of the so called Gray map ϕ : ℤ₄→ℤ₂ ² a relation was established between linear codes over ℤ₄ and certain interesting non-linear binary codes of even length. Here, we shall generalize these notions to codes over ℤ _p2 where p is an arbitrary prime. To this end, a new weight function will be proposed for ℤ _p2 . Further, properties of linear codes over ℤ _p2 will be discussed and the mapping ϕ will be generalized to an isometry between ℤ _p2 and ℤ _p ^p , resp. between ℤ _p2 ⁿ and ℤ _p ^pn . Some properties of Galois rings over ℤ _q will be described and two dual families of linear codes of length n = p ^m − 1, gcd(m, p) = 1, over ℤ _q will be constructed. Taking q = p ², their images under the new mapping can be viewed as a generalization of the binary Kerdock and the Preparata code, although they miss some of their nice combinatorial properties. Received: June 19, 2000; revised version: November 6, 2000     

Specimen Size Requirements for Two-parameter Fracture Toughness Testing

Using a single parameter fracture mechanics theory, a minimum specimen size requirement of min(a, b, B) >200J/σ₀ in tension and min(a, b, B) >25J/σ₀ in bending, where B is the thickness, b the remaining ligament and a is the crack length of the specimen, were derived [Shih and German (1981), International Journal of fracture 17, 27–43] which have provided the basis for modern fracture toughness testing procedures. Two-parameter fracture toughness testing including the constraint, on the other hand, is desirable since it offers a solution to the transferability issue. A size requirement for a valid two-parameter fracture toughness testing based on the J-A₂ three-term solution was determined as min(a, b, B) > 11J/σ₀ [Chao and Zhu (1998), International Journal of fracture 89, 285–307] in which the limiting case is bend specimens under large scale yielding (LSY). Recent work by Chao et al. (2004, International Journal of fracture, 27, 283–302) has shown that the J-A₂ dominance at a crack tip can be significantly enhanced for bending specimens under LSY if a modified J-A₂ solution is adopted. This current paper further studies the size of the J-A₂ dominant zone using the modified J-A₂ solution for deep bend specimens with hardening from low to high and loading from SSY to LSY using finite element analysis. Based on the results, a rather relaxed specimen size requirement min(a, b, B) >6J/σ₀ is developed and recommended for a valid two-parameter fracture toughness testing using the J-A₂ fracture criterion. Validity of the size requirement is demonstrated by using the experimental J-R curves from non-standard bending specimens for A285 steel.     

Periodic Sequences with Maximal Linear Complexity and Large <Emphasis Type="Italic">k</Emphasis>-Error Linear Complexity

The linear complexity and the k-error linear complexity are important concepts for the theory of stream ciphers in cryptology. Keystreams that are suitable for stream ciphers must have large values of these complexity measures. We study periodic sequences over an arbitrary finite field _q and establish conditions under which there are many periodic sequences over _q with period N, maximal linear complexity N, and k-error linear complexity close to N. The existence of many such sequences thwarts attacks against the keystreams by exhaustive search.     

Maximum Distance Separable Convolutional Codes

A maximum distance separable (MDS) block code is a linear code whose distance is maximal among all linear block codes of rate k/n. It is well known that MDS block codes do exist if the field size is more than n. In this paper we generalize this concept to the class of convolutional codes of a fixed rate k/n and a fixed code degree δ. In order to achieve this result we will introduce a natural upper bound for the free distance generalizing the Singleton bound. The main result of the paper shows that this upper bound can be achieved in all cases if one allows sufficiently many field elements. Received: December 10, 1998; revised version: May 14, 1999     

Bounds on the Weight Hierarchies of Extremal Non-chain Codes of Dimension 4

 The weight hierarchy of a linear [n, k; q] code C over GF(q) is the sequence (d ₁, d ₂, . . . , d _k ) where d _r is the smallest support of an r-dimensional subcode of C. An [n, k; q] code is external non-chain if for any r and s, where 1≦r<s≦k, there are no subspaces D and E, such that D⊂E, dim D=r, dim E=s, w _S (D)=d _r , and w _S (E)=d _s . Bounds on the weight hierarchies of such codes of dimension 4 are studied. Received: September 27, 1996     

Esperanza condicionada para probabilidades finitamente aditivas

Let (Ω, θ,J) be a finitely additive probabilistic space formed by any set Ω, an algebra of subsets θ and a finitely additive probabilityJ. In these conditions ifF belongs toV ¹ (Ω, θ,J) there existsf, element of the completion ofL ¹ (Ω, θ,J), such thatF(E)=∫ _E fdJ for allE of θ and conversely. This integral representation gives sense to the following result, which is the objetive of this paper, in terms of the, point function: If β is a subalgebra of θ, for everyF ofV ¹ (Ω, θ,J) there exists a unique element ofV ¹ (Ω, β,J) which we note down byE(F/gb), conditional expetation ofF given β. E(F/β) is characterized by (E(F/β),G)=(F, G) for every ofV ^∞(Ω,β, J). Aside from this, the mappingE(./β):V ¹ (Ω, θ,J)→V ¹ (Ω, β,J) is linear, positive, contractive, idempotent andE(J/β)=J. IfF is ofV ^p (Ω, θ,J),p>1,E(F/β) is ofV ^p (Ω, β,J).

     

Evidence for Dielectric (e–h) and Superconducting (e–e) Pairing in Cuprates

It is demonstrated that anomalous behavior of cuprates can be described in a natural way by the model with partial dielectrization of conduction electron energy spectra (EES). In such a model, near the Fermi surface (FS), there are formed the fours of charged particles (FCP) rather than independent electron–hole (e–h)- and electron–electron (e–e)-pairs. Since the bonding energies are the same for Cooper (e–e)- and dielectric (e–h)-pairs, then the temperature of SC transition T _c due to the Bose-condensation of FCP can be much more than the temperature of Bose-condensation of Cooper pairs only. The dielectric (e–h)-transition is connected with structural phase transformation in the cuprate system, formation of stripe structure, etc. The model successfully describes the presence of maximum at T _c (x)-curve, T-dependence of pseudogap (PG) and SC-gap, effect of nonmagnetic impurities to T _c , STM and ARPES spectra and other properties of cuprates.     

Conductometric Study of Some Metal Halides in Glycerol + Water Mixtures

Electrolytic conductivities of potassium halides, KX (X = Cl⁻, Br⁻, I⁻) have been investigated in 10, 20, and 30 mass% glycerol + H₂O mixtures at 298.0, 308.0, and 318.0 K. The conductance data have been analyzed by the Fuoss-conductance–concentration equation in terms of the limiting molar conductance (Λ⁰), the association constant (K _A ), and the distance of closest approach of ion (R). The association constant (K _A ) tends to increase in the order: 10 mass% < 20 mass% < 30 mass% glycerol + water mixtures, while it decreases with temperature. Thermodynamic parameters ΔH ⁰, ΔG ⁰, and ΔS ⁰ are obtained and discussed. Also, Walden products (Λ⁰η) are reported. The results have been interpreted in terms of ion–solvent interactions and structural changes in the mixed solvents.     

Measurements of Vapor Pressures from 280 to 369 K and (p, ρ, T) Properties from 340 to 400 K at Pressures to 200 MPa for Propane

Measurements of (p, ρ, T) properties for compressed liquid propane have been obtained by means of a metal-bellows variable volumometer at temperatures from 340 to 400 K at pressures up to 200 MPa. The volume- fraction purity of the propane sample was 0.9999. The expanded uncertainties (k = 2) of temperature, pressure, and density measurements have been estimated to be less than 3 mK; 1.5 kPa ( MPa), 0.06% (7 MPa MPa), 0.1% (50 MPa MPa) , and 0.2% (p>150 MPa); and 0.11%, respectively. Four (p, ρ, T) measurements at the same temperatures and pressures as literature values have been conducted for comparisons. In addition, vapor pressures were measured at temperatures from 280 to 369 K. Furthermore, comparisons of available equations of state with the present measurements are reported.Paper presented at the 17th European Conference on Thermophysical Properties, September 5–8, 2005, Bratislava, Slovak Republic.     

New insights into the assessment of k-out-of-n and related systems

k-out-of-n and related systems have received much attention in the recent past years. Hundreds of articles were devoted to various methods to assess them. In this article, we show that there exist very efficient algorithms to compute the reliability of k-out-of-n, l-to-h-out-of-n and consecutive k-out-of-n systems. k-within-r-out-of-n systems are intrinsically much harder. We study the performance of binary decision diagrams (BDDs) on these systems. Then, we propose a new approximation scheme. This algorithm is much more efficient in practice than already proposed methods.     

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     

Extraction of Critical Current Density and Flux Creep Exponent in the Magnetic Superconductor Ru-1212 Using the ac Magnetic Susceptibility Measurements

The ac susceptibility data was employed to extract the temperature dependence of the critical current density, J _c(T), as well as the variation of flux-creep exponent n(T,H _ac) with temperature and ac field amplitude in bulk samples of polycrystalline magnetic superconductor RuSr₂GdCu₂O₈ (Ru-1212). The critical state models and the collective flux-creep approximation model were successfully accounted to describe such behavior below the transition temperature. The calculated values of n(T,H) are well fitted to a power law of the following form: n(T,H)=n ₀(H)T ^−s(H), where s is field dependent exponent whose values varied from −2.4, −1.01 for field amplitudes ranging from 0.5 G and 3.8 G. The power law describing the frequency dependence of χ′ is found to be consistent with the results of the current-dependent effective activation energy of the form U(J)=U ₀ln (J _c/J). Additionally, the dependence of the current density is found to scale according as: J _c(T)=J _c0(1−T/T _c) ⁿ , where the exponent n values varied from 1.05 to 1.25. Such dependence is an indication of intergrain coupling that could be ascribed in terms of superconductor–insulator–superconductor junctions. The derived temperature dependence of J _c(T) is in good agreement with the data obtained from the measurements using the traditional “loss-maximum” approach. Furthermore, the flux-creep effect increased with increasing both ac fields and temperatures except at about 15–25 K below the onset of T _c, where a slowing down of the flux creep was observed.