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Boundedwidth polynomialsize branching programs recognize exactly those languages
 in NC’, in “Proceedings, 18th ACM STOC
, 1986
"... We show that any language recognized by an NC ’ circuit (fanin 2, depth O(log n)) can be recognized by a width5 polynomialsize branching program. As any boundedwidth polynomialsize branching program can be simulated by an NC ’ circuit, we have that the class of languages recognized by such prog ..."
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Cited by 209 (13 self)
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We show that any language recognized by an NC ’ circuit (fanin 2, depth O(log n)) can be recognized by a width5 polynomialsize branching program. As any boundedwidth polynomialsize branching program can be simulated by an NC ’ circuit, we have that the class of languages recognized by such programs is exactly nonuniform NC’. Further, following
Upper Bounds for ConstantWeight Codes
 IEEE TRANS. INFORM. THEORY
, 2000
"... Let A(n; d; w) denote the maximum possible number of codewords in an (n; d; w) constantweight binary code. We improve upon the best known upper bounds on A(n; d; w) in numerous instances for n 6 24 and d 6 12, which is the parameter range of existing tables. Most improvements occur for d = 8; 10, ..."
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Cited by 31 (1 self)
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Let A(n; d; w) denote the maximum possible number of codewords in an (n; d; w) constantweight binary code. We improve upon the best known upper bounds on A(n; d; w) in numerous instances for n 6 24 and d 6 12, which is the parameter range of existing tables. Most improvements occur for d = 8; 10, where we reduce the upper bounds in more than half of the unresolved cases. We also extend the existing tables up to n 6 28 and d 6 14. To obtain these results, we develop new techniques and introduce new classes of codes. We derive a number of general bounds on A(n; d; w) by means of mapping constantweight codes into Euclidean space. This approach produces, among other results, a bound on A(n; d; w) that is tighter than the Johnson bound. A similar improvement over the best known bounds for doublyconstantweight codes, studied by Johnson and Levenshtein, is obtained in the same way. Furthermore, we introduce the concept of doublyboundedweight codes, which may be thought of as a generaliz...
Lower bounds on trellis complexity of block codes
 IEEE 2hns. Inform. Theory
, 1995
"... AbstructThe trellis statecomplexity s of a linear block code is defined as the logarithm of the maximum number of states in its minimal trellis. We present a new lower bound on the statecomplexity of linear codes, which includes most of the existing bounds as special cases. The new bound is obtain ..."
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Cited by 22 (5 self)
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AbstructThe trellis statecomplexity s of a linear block code is defined as the logarithm of the maximum number of states in its minimal trellis. We present a new lower bound on the statecomplexity of linear codes, which includes most of the existing bounds as special cases. The new bound is obtained by dividing the time axis for the code into several sections of varying lengths, as opposed to the division into two sectionsthe past and the futureemployed in the wellknown DLP bounds. For a large number of codes this results in a considerable improvement upon the DLP bound. Moreover, we generalize the new bound to nonlinear codes, and introduce several alternative techniques for lowerbounding the trellis complexity, based on the distance spectrum and other combinatorial properties of the code. We also show how the general ideas developed in this paper may be employed to lowerbound the maximum and the total number of branches in the trellis, leading to considerably tighter bounds on these quantities. Furthermore, the asymptotic behavior of the new bounds is investigated, and shown to improve upon the previously known asymptotic estimates of trellis statecomplexity. Index Terms Trellis complexity, block codes, maximumlikelihood decoding, generalized Hamming weight hierarchy, dynamics of codes. E I.
The Riemann problem near a hyperbolic singularity
 II, SIAM J. Appl. Math., this issue
"... Abstract. This paper is interested in classifying the solutions of Riemann problems for the 2 x 2 conservation laws that have homogeneous quadratic flux functions. Such flux functions approximate an arbitrary 2 x 2 system in a neighborhood of an isolated point where strict hyperbolicity fails. Here ..."
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Cited by 17 (9 self)
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Abstract. This paper is interested in classifying the solutions of Riemann problems for the 2 x 2 conservation laws that have homogeneous quadratic flux functions. Such flux functions approximate an arbitrary 2 x 2 system in a neighborhood of an isolated point where strict hyperbolicity fails. Here the solution for the symmetric systems in Region II of the four region classification of Schaeffer and Shearer is given. The solution is based on the qualitative shape of the integral curves described by Schaeffer and Shearer and a numerical calculation of the Hugoniot loci and their shock types. Key words. Riemann problem, nonstrictly hyperbolic conservation laws, umbilic points AMS(MOS) subject classifications. 65M10, 76N99, 35L65, 35L67
Nonclassical shocks and kinetic relations: finite difference schemes
 SIAM J. Numer. Anal
, 1998
"... This paper is dedicated to the memory of Ami Harten, who, along with coinvestigators Mac Hyman and Peter Lax, was among the avant garde in the study of entropyviolating shocks in numerical schemes. Abstract. We consider hyperbolic systems of conservation laws that are not genuinely nonlinear. The s ..."
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Cited by 15 (4 self)
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This paper is dedicated to the memory of Ami Harten, who, along with coinvestigators Mac Hyman and Peter Lax, was among the avant garde in the study of entropyviolating shocks in numerical schemes. Abstract. We consider hyperbolic systems of conservation laws that are not genuinely nonlinear. The solutions generated by diffusivedispersive regularizations may include nonclassical (n.c.) shock waves that do not satisfy the classical Liu entropy criterion. We investigate the numerical approximation of n.c. shocks via conservative difference schemes constrained only by a single entropy inequality. The schemes are designed by comparing their equivalent equations with the continuous model and include discretizations of the diffusive and dispersive terms. Limits of these schemes are characterized via the kinetic relation introduced earlier by the authors. We determine the kinetic function numerically for several examples of systems and schemes. This study demonstrates that the kinetic relation is a suitable tool for the selection of unique n.c. solutions and for the study of their sensitive dependence on the critical parameters: the ratios of diffusion/dispersion and diffusion/mesh size, the shock strength, and the order of discretization of the flux.