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181
The Z_4linearity of Kerdock, Preparata, Goethals, and related codes
, 2001
"... Certain notorious nonlinear binary codes contain more codewords than any known linear code. These include the codes constructed by NordstromRobinson, Kerdock, Preparata, Goethals, and DelsarteGoethals. It is shown here that all these codes can be very simply constructed as binary images under the ..."
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Cited by 107 (15 self)
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Certain notorious nonlinear binary codes contain more codewords than any known linear code. These include the codes constructed by NordstromRobinson, Kerdock, Preparata, Goethals, and DelsarteGoethals. It is shown here that all these codes can be very simply constructed as binary images under the Gray map of linear codes over ¡ 4, the integers mod 4 (although this requires a slight modification of the Preparata and Goethals codes). The construction implies that all these binary codes are distance invariant. Duality in the ¡ 4 domain implies that the binary images have dual weight distributions. The Kerdock and ‘Preparata ’ codes are duals over ¡ 4 — and the NordstromRobinson code is selfdual — which explains why their weight distributions are dual to each other. The Kerdock and ‘Preparata ’ codes are ¡ 4analogues of firstorder ReedMuller and extended Hamming codes, respectively. All these codes are extended cyclic codes over ¡ 4, which greatly simplifies encoding and decoding. An algebraic harddecision decoding algorithm is given for the ‘Preparata ’ code and a Hadamardtransform softdecision decoding algorithm for the Kerdock code. Binary first and secondorder ReedMuller codes are also linear over ¡ 4, but extended Hamming codes of length n ≥ 32 and the
Semidefinite Programming and Combinatorial Optimization
 DOC. MATH. J. DMV
, 1998
"... We describe a few applications of semide nite programming in combinatorial optimization. ..."
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Cited by 99 (1 self)
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We describe a few applications of semide nite programming in combinatorial optimization.
Landscapes and Their Correlation Functions
, 1996
"... Fitness landscapes are an important concept in molecular evolution. Many important examples of landscapes in physics and combinatorial optimation, which are widely used as model landscapes in simulations of molecular evolution and adaptation, are "elementary", i.e., they are (up to an additive const ..."
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Cited by 89 (15 self)
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Fitness landscapes are an important concept in molecular evolution. Many important examples of landscapes in physics and combinatorial optimation, which are widely used as model landscapes in simulations of molecular evolution and adaptation, are "elementary", i.e., they are (up to an additive constant) eigenfuctions of a graph Laplacian. It is shown that elementary landscapes are characterized by their correlation functions. The correlation functions are in turn uniquely determined by the geometry of the underlying configuration space and the nearest neighbor correlation of the elementary landscape. Two types of correlation functions are investigated here: the correlation of a time series sampled along a random walk on the landscape and the correlation function with respect to a partition of the set of all vertex pairs.
On Metric RamseyType Phenomena
"... The main question studied in this article may be viewed as a nonlinear analog of Dvoretzky's Theorem in Banach space theory or as part of Ramsey Theory in combinatorics. ..."
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Cited by 69 (39 self)
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The main question studied in this article may be viewed as a nonlinear analog of Dvoretzky's Theorem in Banach space theory or as part of Ramsey Theory in combinatorics.
Semidefinite Programming and Integer Programming
"... We survey how semidefinite programming can be used for finding good approximative solutions to hard combinatorial optimization problems. ..."
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Cited by 48 (7 self)
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We survey how semidefinite programming can be used for finding good approximative solutions to hard combinatorial optimization problems.
The geometry of twoweight codes
 BULLETIN OF THE LONDON MATHEMATICAL SOCIETY
, 1986
"... We survey the relationships between twoweight linear [n, k] codes over GF(q), projective (n, k, h1, h2,) sets in PG(k — 1,q), and certain strongly regular graphs. We also describe and tabulate essentially all the known examples. ..."
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Cited by 40 (1 self)
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We survey the relationships between twoweight linear [n, k] codes over GF(q), projective (n, k, h1, h2,) sets in PG(k — 1,q), and certain strongly regular graphs. We also describe and tabulate essentially all the known examples.
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...
New Upper Bounds on Error Exponents
"... We derive new upper bounds on the error exponents for the maximum likelihood decoding and error detecting in the binary symmetric channels. This is an improvement on the straightline bound by ShannonGallagerBerlekamp (1967) and the McElieceOmura (1977) minimum distance bound. For the probability ..."
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Cited by 28 (6 self)
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We derive new upper bounds on the error exponents for the maximum likelihood decoding and error detecting in the binary symmetric channels. This is an improvement on the straightline bound by ShannonGallagerBerlekamp (1967) and the McElieceOmura (1977) minimum distance bound. For the probability of undetected error the new bounds are better than the recent bound by AbdelGhaffar (1997) and the minimum distance and straightline bounds by Levenshtein (1978, 1989). We further extend the range of rates where the undetected error exponent is known to be exact. Keywords: Error exponents, Undetected error, Maximum likelihood decoding, Distance distribution, Krawtchouk polynomials. Submitted to IEEE Transactions on Information Theory 1 Introduction A classical problem of the information theory is to estimate probabilities of undetected and decoding errors when a block code is used for information transmission over a binary symmetric channel (BSC). We will study here exponential bounds ...
New upper bounds for kissing numbers from semidefinite programming
 Journal of the American Mathematical Society
, 2006
"... In geometry, the kissing number problem asks for the maximum number τn of unit spheres that can simultaneously touch the unit sphere in ndimensional Euclidean space without pairwise overlapping. The value of τn is only known for n =1, 2, 3, 4, 8, 24. While its determination for n =1, 2 is trivial, ..."
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Cited by 28 (13 self)
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In geometry, the kissing number problem asks for the maximum number τn of unit spheres that can simultaneously touch the unit sphere in ndimensional Euclidean space without pairwise overlapping. The value of τn is only known for n =1, 2, 3, 4, 8, 24. While its determination for n =1, 2 is trivial, it is not the case