Projective Systems Whose Support is the Union of Two Linear Subspaces with Nonempty Intersection

We consider the code corresponding to a projective system whose support is the union of two linear subspaces with nonempty intersection, and prove that the code is uniquely determined up to equivalence by its weight enumerator. Also we obtain the same conclusion for the complement of the union of two linear subspaces.The authors are supported by Korea Research Foundation Grant (KRF-2001-005-D00002).     

A New Class of Ternary Cocyclic Hadamard Codes

 We present a new construction for p-ary codes meeting the Plotkin bound, for any odd prime p, from any planar function on the additive group of GF(p ^a ). We use the Coulter-Matthews planar functions with p=3 to construct new families of ternary cocyclic codes, and compute their dimensions for a≤6. Received: May 23, 2002; revised version: March 27, 2003 Keywords: Plotkin bound, Cocyclic code, Hadamard code, Planar function.     

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     

The Matroid of Supports of A Linear Code

 A relation between the Hamming weight enumerator of a linear code and the Tutte polynomial of the corresponding matroid has been known since long ago. It provides a simple proof of the MacWilliams equation (see D. Welsh, Matroid Theory (1976)). In this paper we prove analogous results for the support weight distributions of a code. Received March 5, 1996; revised version October 28, 1996     

Maximal AMDS codes

Complete (n, k)-arcs in PG(k − 1, q) and projective (n, k) _q -AMDS codes that admit no projective extensions are equivalent objects. We show that projective AMDS codes of reasonable length admit only linear extensions. Thus, we are able to prove the maximality of many known linear AMDS codes. At the same time our results sharply limit the possibilities for constructing long nonlinear AMDS codes. We also show that certain short linear AMDS codes are maximal. Central to our approach is the Bruen–Silverman model of linear codes first introduced in Alderson (On MDS codes and Bruen–Silverman codes. Ph.D. Thesis, University of Western Ontario, 2002) and Alderson et al. (J. Combin. Theory Ser. A 114(6), 1101–1117, 2007). The authors acknowledge support from the N.S.E.R.C. of Canada.     

Explicit expressions of eigenvalues and eigenvectors for transversely isotropic piezoelectric materials

Summary.  Explicit expressions of eigenvalues and associated eigenvectors in Stroh's theory [1] for transversely isotropic piezoelectric materials under generalized plane deformation are presented. It is shown that the practical calculation for some piezoelectric ceramics yields all eigenvalues which are distinct from each other for generalized x-z plane deformation, where z is the poling axis of a transversely isotropic piezoelectric material. Some numerical results based on the explicit expressions are given, and the orthogonality and closed relation between matrices A and B in Stroh's theory [2], [3] for piezoelectric material are verified numerically. Received May 8, 2002; revised December 14, 2002 Published online: May 20, 2003 The paper is supported by the doctorate foundation of Xi'an Jiaotong University. The science foundation of the Shaanxi Province of China is appreciated, too.     

A Polynomial Representation of the Diffie-Hellman Mapping

 Let F _q be the finite field of order q and γ be an element of F _q of order d. The construction of an explicit polynomial of degree with the property for is described. In particular the exact degree and sparsity of f are determined. Received: February 8, 2001; revised version: August 8, 2002 Keywords: Diffie-Hellman cryptosystem, Polynomial interpolation, Degree, Sparsity. The work was partially supported by the Austrian Science Fund (FWF) under the project S8306-MAT     

Toric Varieties Hirzebruch Surfaces and Error-Correcting Codes

 For any integral convex polytope in ℝ there is an explicit construction of an error-correcting code of length (q-1)² over the finite field 𝔽 _q , obtained by evaluation of rational functions on a toric surface associated to the polytope. The dimension of the code is equal to the number of integral points in the given polytope and the minimum distance is determined using the cohomology and intersection theory of the underlying surfaces. In detail we treat Hirzebruch surfaces. Received: August 21, 2000; revised version: September 3, 2002     

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     

Resolvent Representation for Regular Differential Ideals

 We show that the generic zeros of a differential ideal [A]:H ^∞ _A defined by a differential chain A are birationally equivalent to the general zeros of a single regular differential polynomial. This provides a generalization of both the cyclic vector construction for system of linear differential equations and the rational univariate representation of algebraic zero dimensional radical ideals. In order to achieve generality, we prove new results on differential dimension and relative orders which are of independent interest. Received: June 13, 2001; revised version: May 2, 2002 Key words: Differential algebra, Differential primitive element, Cyclic vector, Computer algebra, Resolvent.     

A decomposition of the class of semiconvex games

Summary The paper studies the classSC ^N of cooperative games with player setN which have the semiconvexity property.SC ^N is decomposed into an algebraic sum of convex cones of games for which generating sets are available. The union of these sets thus forms a generating set forSC ^N . Special attention is paid to one of the considered cones, in the decomposition ofSC ^N . In particular, the so called airport savings gamesW _y, ℝ ^N , defined byw _y (S)=Σ _jεS Y _i −max _jεS Y _j for 0 ≠S⋞N, are emphasized. This work is part of the Ph.D. thesis of the first author (Derks 1991).     

Solvable Brioschi Resolvents

 The Brioschi resolvent y ⁵ −10Zy ³ +45Z ² y−Z ² is a key component of an algorithm for calculating the roots of a general quintic polynomial. It is obtained from the general quintic polynomial by applying two Tschirnhausen transformations. In this paper it is shown that if the quintic polynomial is a solvable polynomial, then its associated parameter Z in the Brioschi resolvent satisfies Z=g(t) where g(t) is a rational function in ℚ(t) and t is chosen from an appropriate field. Received: July 23, 2001; revised version: October 14, 2002 Keywords: Solvable quintic, Brioschi quintic, resolvent sextic, parametrization. The author was supported by a research grant from the Natural Sciences and Engineering Research Council of Canada     

The Minimal Generator Matrix of a Vector Space

The atomic vectors of a finitely generated vector space C over a field F are characterized for C a subspace of the product vector space ? = ∏ _{i =1} ⁿ ? _i over F. For finite fields, the minimal trellis diagram for mixed-codes is determined, and this provides the L-section minimal trellis diagram for linear codes. As an example, an extremely simple yet comprehensive analysis of the trellis structure of Reed-Muller codes is given. In particular, a trellis oriented generator matrix for the 2 ^l -section minimal trellis diagram of a Reed-Muller code is presented. Received: February 27, 1997; revised version: May 6, 1999     

DFT Domain Characterization of Quasi-Cyclic Codes

 The transform domain characterization of linear cyclic codes over finite fields using Discrete Fourier Transform (DFT) over an appropriate extension field is well known. In this paper, we extend this transform domain characterization for linear quasi-cyclic codes over finite fields. We show how one can derive a lower bound on the minimum Hamming distance of a quasi-cyclic code and decode the code upto that minimum Hamming distance using this characterization. Received: January 17, 2002; revised version: November 30, 2002 RID="*" ID="*" This work was partly supported by CSIR, India, through Research Grant (22(0298)/99/EMR-II) to B. S. Rajan Keywords: Quasi-cyclic codes, Discrete Fourier transform, Cyclotomic cosets. Part of this work was presented in ICCCD 2000, Kharagpur, India and ISIT 2001, Washington D.C., USA     

Preparation of eutectics by directional solidification of quaternary melts

This paper presents a theoretical analysis of quasi-equilibrium directional solidification of a quaternary melt. We consider the variation in the composition of phases in each portion of the sample and changes in phase composition for various types of phase reactions. The results indicate that the melt trajectory during directional solidification may belong only to those phase diagram elements corresponding to the crystallization of binary, ternary, or quaternary eutectics or single-phase crystallization regions. Using the directional solidification of a melt with the composition (at %) Cu 29.37, Ni 17.72, Fe 5.91, and S 47.00, we obtained a sample consisting of zones with different phase compositions: [Ni _z (Fe,Cu)_{1 − z} ]S_{1 ± δ} single-phase zone, [Ni _z (Fe,Cu)_{1 − z} ]S_{1 ± δ} + Cu_{5 ± x} Fe_{1 ± x} S₄ binary eutectic mixture, and [Ni _z (Fe,Cu)_{1 − z} ]S_{1 ± δ} + Cu_{5 ± x} Fe_{1 ± x} S₄ + (Ni _z Fe_{1 − z} )S₂ ternary eutectic mixture. In going from one zone to another, new phases appear and the average composition of the sample changes sharply, whereas the compositions of the melt and solid solution present in neighboring zones vary continuously. These results are consistent with theoretical concepts.     

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     

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     

Fast Parallel Algorithms for Matrix Reduction to Normal Forms

 We investigate fast parallel algorithms to compute normal forms of matrices and the corresponding transformations. Given a matrix B in ℳ _n,n (K), where K is an arbitrary commutative field, we establish that computing a similarity transformation P such that F=P ^-1 BP is in Frobenius normal form can be done in ?C ² _K . Using a reduction to this first problem, a similar fact is then proved for the Smith normal form S(x) of a polynomial matrix A(x) in ℳ _n,m (K[x]); to compute unimodular matrices U(x) and V(x) such that S(x)=U(x)A(x)V(x) can be done in ?C ² _K . We get that over concrete fields such as the rationals, these problems are in ?C ². Using our previous results we have thus established that the problems of computing transformations over a field extension for the Jordan normal form, and transformations over the input field for the Frobenius and the Smith normal form are all in ?C ² _K . As a corollary we establish a polynomial-time sequential algorithm to compute transformations for the Smith form over K[x]. Received: February 29, 1996; revised version: August 29, 1997     

Mechanistic investigation of the field-activated combustion synthesis (FACS) of tungsten carbide with or without nickel additive

The activation of self-propagating combustion reactions in the W-C system and its composite with Ni as additive was achieved by using an electric field. The reaction mechanisms of Field-Activated Combustion Synthesis (FACS) of tungsten carbide and its composite have been investigated by using sample-quenched method. Through turning off electric field during FACS process, a series of combustion products with different phase compositions have been obtained. Layer to layer X-ray and microscopic analyses of these combustion products across quenched combustion front suggested that the synthesis of WC is a process involving the solid diffusion of carbon into a carbide layer. W₂C is the intermediate phase between WC and reactants (W and C). Metal additive produces liquid phase and accelerates the diffusion between solid reactants (W and C), which facilitates the formation of W₂C and the transformation from W₂C to WC phase. Moreover, melted Ni reacts with W and W₂C to form mixed compounds of type W _x C _y M _z .     

Mixed-State Specific Heat of the Type-II Superconductor Nb0.77Zr0.23 in Magnetic Fields up toB c2

In order to document the behavior of the mean-field mixed-state specific heat of an isotropic. strongly type-II superconductor (i.e., with a large value of the Ginzburg parameterk), and to provide a basis for comparison with high-temperature superconductors, we measured the specific heatC of the alloy Nb_0.77Zr_0.23 withT _c = 10.8K, B _c2 (0) = 7.9T, in magnetic fieldsB = 0, 0.2, 1.0, 12, 2.0, 2.4. 3.0, 3.3. 4.0. 4.4, 4.8, 5.2, 6.0. 6.6, 7.2 and 10 T. The values of the upper critical fieldB _c2 ( T), thermodynamic critical fieldB _c (T), Ginzburg parameterk(T), and coefficient γ(B) = lim_{T0(C(T. B)/T)} are derived from the specific heat data and found to be in agreement with the GLAG theory in the dirty limit. The behavior of the mixed-state specific heat is analyzed in terms ofC _el /T,∂(C _el /T)/∂B, and∂(C _el /T)/∂T vs. Tcurves, whereC _el is the electronic contribution to the specific heat.