Long binary narrow-sense BCH codes are normal

Let C be the binary narrow-sense BCH code of length n = (2 ^m − l)/h, where m is the order of 2 modulo n. Using characters of finite fields and a theorem of Weil, and results of Vladut-Skorobogatov and Lang-Weil we prove that the code C is normal in the non-primitive case h > 1 if 2^m ≥ 4(2th)^{4t + 2}, and in the primitive case h = 1 if m ≥ m ₀ where the constant m ₀ depends only on t.     

The minimum distance of parameterized codes on projective tori

Let X be a subset of a projective space, over a finite field K, which is parameterized by the monomials arising from the edges of a clutter. Let I(X) be the vanishing ideal of X. It is shown that I(X) is a complete intersection if and only if X is a projective torus. In this case we determine the minimum distance of any parameterized linear code arising from X.     

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 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.     

Hyper-Kloosterman Sums and their Applications to the Coding Theory

The dual code of the Melas code is called the Kloosterman code. The weights of its codewords can be expressed by the Kloosterman sums, and are uniformly distributed with respect to the Sato-Tate measure. In this paper, the hyper-Kloosterman code C _m (q), a generalization of the Kloosterman code is defined, and the uniform distribution property is deduced using the hyper-Kloosterman sums when m is even and p− 1|m. Finally we discuss doubly-evenness for the weights in the binary case. It shows that we can construct infinitely many doubly-even codes in a non-trivial way. Received: October 30, 2000; revised version: February 28, 2001     

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     

Chryseobacterium meningosepticum in a parapneumonic effusion in a chronic kidney disease patient on hemodialysis

Chryseobacterium meningosepticum is an uncommon human pathogen, which is an inhabitant of soil and water. It should be included in the list of suspected nosocomial infections, especially in patients with immunocompromised status. C. meningosepticum infections are not common but are clinically important because the organism is naturally resistant to multiple antibiotics. We report a case where the bacterium was isolated from the pleural fluid from a chronic kidney disease patient on hemodialysis, who developed pneumonia with complicated parapneumonic effusion. To the best of our knowedge, this is the first case where C. meningosepticum is isolated from the pleural effusion, from India.     

Evaluation of nuclear safety from the outputs of computer codes in the presence of uncertainties

We apply methods from order statistics to the problem of satisfying regulations that specify individual criteria to be met by each of a number of outputs, k, from a computer code simulating nuclear accidents. The regulations are assumed to apply to an ‘extent’, γ_k, (such as 95%) of the cumulative probability distribution of each output, k, that is obtained by randomly varying the inputs to the code over their ranges of uncertainty. We use a ‘bracketing’ approach to obtain expressions for the confidence, β, or probability that these desired extents will be covered in N runs of the code. Detailed results are obtained for k=1,2,3, with equal extents, γ, and are shown to depend on the degree of correlation of the outputs. They reduce to the proper expressions in limiting cases. These limiting cases are also analyzed for an arbitrary number of outputs, k.The bracketing methodology is contrasted with the traditional ‘coverage’ approach in which the objective is to obtain a range of outputs that enclose a total fraction, γ, of all possible outputs, without regard to the extent of individual outputs. For the case of two outputs we develop an alternate formulation and show that the confidence, β, depends on the degree of correlation between outputs. The alternate formulation reduces to the single output case when the outputs are so well correlated that the coverage criterion is always met in a single run of the code if either output lies beyond an extent γ, it reduces to Wilks' expression for un-correlated variables when the outputs are independent, and it reduces to Wald's result when the outputs are so negatively correlated that the coverage criterion could never be met by the two outputs of a single run of the code. The predictions of both formulations are validated by comparison with Monte Carlo simulations.     

Constraints on Weights in Binary Codes

 There are known constraints on the number of doubly-even and singly-even vectors in a linear even binary code C. These constraints give information about C∩C ^⊥. We find new constraints on weights in a binary linear code C which also contains odd weight vectors. This leads to information about the dimension and weights of C∩C ^⊥. Received: September 20, 1996     

Adherence properties of metal-to-ceramic joints

A study is reported of the fracture mechanics of metal-to-ceramic laminates (Al₂O₃/Nb, Si₃N₄/Zr) produced by solid-state bonding. The bond quality of notched bend and tension specimens is described in terms of a stress intensity factor K _ICV, which is evaluated in the same way as for isotropic material. K _ICV is measured as functions of the microstructure of the constituents, the metal layer thickness, the environmental conditions and the test temperature. At very high test temperatures where plastic flow occurs, the J-integral was used for bond quality characterization. For the linear elastic case, the J-value is compared with the stress intensity concept K _ICV. In contrast to the behaviour of a bulk ceramic material, the J-value increases with increasing test temperature if a thick metal layer (3 mm) is used.     

The dependence of broadband four-wave mixing signal intensity on the length of the interaction region

Abstract

A theoretical investigation of broadband four-wave mixing (FWM) signal intensity dependence on the length of the interaction region, L, is undertaken, for the case of weak intensity input beams. The phase diffusing model, PDM, is used to describe the statistics of the field fluctuations. A general result is derived which is in agreement with the quadratic L dependence obtained in previous work by Cooper et al. 1989, Phys. Rev. A, 40, 5707, where only the particular case of coherent lengths, L _c , much greater than L was considered. We particularize our general result for the opposite case, L _c ?L, where retardation effects in the randomly fluctuating phases can no longer be neglected. In this case we find the FWM signal intensity to have a linear, rather than a quadratic, dependence on L. An intuitive and simple physical justification is provided for this new result in terms of the short length scale, coherent dipole, contributions to the FWM signal, along the interaction region.     

Non-degenerate Linear Codes Determined by their Weight Enumerators

 In [1], Homma-Kim-Yoo showed that a code C such that the support of the associated projective system is a union of two linear subspaces, is uniquely determined up to equivalence by its weight enumerator. Let W _C (z) be the weight enumerator of a code C. Assuming certain conditions on the coefficients of W _C (z) for some range, we prove that the support of the associated projective system is a union of two linear subspaces. Received: February 27, 2002; revised version: October 2, 2002 Keywords: Linear code, Projective system, Weight enumerator. MSC2000: 94B27, 94B65, 51E20 Partially supported by Grant-in-Aid for Scientific Research, Ministry of Education, #12640180     

Travelling waves in autocatalytic chemical systems with decay: bounds on existence

The existence of solutions to the travelling-wave equations governed by an autocatalytic reaction of order p (p ≥ 1) and an autocatalytic decay step of order q (q ≥ 1) are examined in the limit of large p. Two cases are treated, q of O(1) and . In the first case, an upper bound k _crit is found for k for the existence of a solution, where k is a dimensionless measure of the strength of the decay step. In the second case, an upper bound on k is also found when q < p. For q ≥ p, there is no upper bound on k and solutions exist for all (positive) values of k.     

Construction of binary minimal product parity-check matrices

A parity-check matrix H of a given code is called minimal if it has minimum number of nonzero entries among all parity-check matrices representing . Let and be two binary linear block codes with minimal parity-check matrices H ₁ and H ₂, respectively. It is shown that, using H ₁ and H ₂, one can efficiently generate a minimal parity-check matrix for the product code .     

DEM Simulations of Toner Particles with an O (N log N) Hierarchical Tree Code Algorithm

The computation time for a particle simulation where long-reaching forces, such as electrostatic forces for charged particles, have to be taken into account, rises asymptotically with O (N ²). A significant reduction of computation time can be achieved by applying a tree code algorithm. The algorithm exploits the idea of grouping suitable particles to macroparticles thus reducing the number of necessary calculations within a reasonably limited error bound. Although this algorithm was developed for astronomical galaxy simulations, it can be also used for simulations of microscopical particle with finite radii. The tree code can be used for an efficient collision routine if some additional boundary conditions are taken into account. This results in an asymptotic computation time dependence of O (N log N).     

Algebraic decoding of the binary systematic (71, 36, 11) quadratic residue code

A fast and efficient algebraic decoding algorithm (ADA) is proposed to correct up to five possible error patterns in the binary systematic (71,?36,?11) quadratic residue (QR) code. The technique required here is based on the ADA developed by He et al. [2001. Decoding the (47,?24,?11) quadratic residue code. IEEE transactions on information theory, 47 (3), 1181–1186.] and the modification of the ADA developed by Lin et al. [2010. Decoding of the (31,?16,?7) quadratic residue code. Journal of the Chinese institute of engineers, 33 (4), 573–580; 2010. High speed decoding of the binary (47,?24,?11) quadratic residue code. Information sciences, 180 (20), 4060–4068]. The new proposed conditions and the error-locator polynomials for different numbers of errors in the received word are derived. Simulation results show that the decoding speed of the proposed ADA is faster than the other existing ADAs.     

New statistical distributions for group counting in?Bernoulli and Poisson processes

Based on a sequential group sampling procedure, the authors introduce a new statistical distribution for group-by-group counting for Bernoulli processes. Suppose that a population contains an infinite number of items, each item having probability p for nonconformity and probability 1−p for conformity. Items are inspected group by group, where each group contains m items, and the number of nonconforming items are recorded only when inspection of a group is completed. The inspection procedure is terminated if the cumulative number of nonconforming items recorded is greater than or equal to a specified number r. The distribution introduced in this paper consists of two random variables: the number of groups inspected, K, and the cumulative number of nonconforming items observed, X. Statistical properties and inferences for the bivariate distribution of (K,X) are studied. The authors also discuss the limiting case as m→∞ and p→0 with mp fixed, and derive a new bivariate distribution for group-by-group counting for Poisson processes that approaches the gamma distribution in limit. A new type of control chart and an acceptance sampling procedure are developed based on (K,X), and examples are given to illustrate their applications.     

High strain rate superplasticity in powder metallurgy aluminium alloy 6061 + 20 vol.-%SiCpcomposite with relatively large particle size

Abstract

The possibility of high strain rate superplasticity (HSRS) was examined over a wide range of temperatures in a powder metallurgy aluminium alloy 6061/SiC_p composite with a relatively large SiC particle size of ~8 μm. A maximum tensile elongation of 350% was obtained at 600°C and 10^-2 s^-1. Tensile elongations over 200% were obtained in a narrow temperature range between 590 and 610°C at high strain rates of 10^-2 and 10^-1 s^-1. The current testing temperature range could be divided into two regions depending on the rate-controlling deformation mechanism. Region I is in the lower temperature range from 430 to 490°C, where lattice diffusion controlled dislocation climb creep (n = 5) is the rate-controlling deformation process, and region II is in the higher temperature range from 520 to 610°C, where lattice diffusion controlled grain boundary sliding controls the plastic flow. An abnormally large increase in activation energy was noted at temperatures above 590°C, where large tensile elonga tions over 200% were obtained at high strain rates. This increase in activation energy and high tensile ductility may be explained in terms of presence of a liquid phase created by partial melting, but such evidence could not be provided by the current differential scanning calorimetry (DSC) test. This may be because the DSC is not sensitive enough to detect the small amount of liquid phase.     

On the Equivalence of Codes over Finite Rings

It is known that if a finite ring R is Frobenius then equivalences of linear codes over R are always monomial transformations. Among other results, in this paper we show that the converse of this result holds for finite local and homogeneous semilocal rings. Namely, it is shown that for every finite ring R which is a direct sum of local and homogeneous semilocal subrings, if every Hamming-weight preserving R-linear transformation of a codeC₁ onto a code C₂ is a monomial transformation then R is a Frobenius ring.Partially supported by Center of Ring Theory and Applications, Ohio University.