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We apply an improvement of the Delsarte LP-bound to give a new proof of the non-existence of finite projective planes of order 6, and uniqueness of finite projective planes of order 7. The proof is computer aided, and it is also f...
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We apply an improvement of the Delsarte LP-bound to give a new proof of the non-existence of finite projective planes of order 6, and uniqueness of finite projective planes of order 7. The proof is computer aided, and it is also feasible to apply to higher orders like 8, 9 and, with further improvements, possibly 10 and 12.
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In this paper the problem for improvement of the Delsarte bound for τ -designs is investigated. Two main results are presented. Firstly, necessary and sufficient conditions for improving the bound are proved. We define test funct...
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In this paper the problem for improvement of the Delsarte bound for τ -designs is investigated. Two main results are presented. Firstly, necessary and sufficient conditions for improving the bound are proved. We define test functions with the property that they are negative if and only if the Delsarte bound D(M, τ) can be improved by linear programming. Then we investigate the infinite polynomial metric spaces and give exact intervals, when the Delsarte bound is not the best linear programming bound possible. Secondly, we derive a new bound for the infinite PMS. Analytical forms of the extremal polynomials of degree τ + 2 for non-antipodal PMS and of degree τ + 3 for antipodal PMS are given. The new bound is investigated in different asymptotical processes for infinite PMS. When τ and n grow simultaneously to infinity our bound is better than Delsarte bound.
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Association schemes have many applications to the study of designs, codes, and geometries and are well studied. Coherent configurations are a natural generalization of association schemes, however, analogous applications have yet ...
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Association schemes have many applications to the study of designs, codes, and geometries and are well studied. Coherent configurations are a natural generalization of association schemes, however, analogous applications have yet to be fully explored. Recently, Hobart [Mich. Math. J. 58:231-239, 2009] generalized the linear programming bound for association schemes, showing that a subset Y of a coherent configuration determines positive semidefinite matrices, which can be used to constrain certain properties of the subset. The bounds are tight when one of these matrices is singular, and in this paper we show that this gives information on the relations between Y and any other subset. We apply this result to sets of nonincident points and lines in coherent configurations determined by projective planes (where the points of the subset correspond to a maximal arc) and partial geometries.
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For nonnegative integers q, n, d, let A(q) (n, d) denote the maximum cardinality of a code of length n over an alphabet [q] with q letters and with minimum distance at least d. We consider the following upper bound on A(q) (n, d)....
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For nonnegative integers q, n, d, let A(q) (n, d) denote the maximum cardinality of a code of length n over an alphabet [q] with q letters and with minimum distance at least d. We consider the following upper bound on A(q) (n, d). For any k, let C-k be the collection of codes of cardinality at most k. Then A(q) (n, d) is at most the maximum value of Sigma(n)(upsilon is an element of[q]) x({upsilon}), where x is a function C-4 -> R+ such that x(empty set) = 1 and x(C) = 0 if C hasminimum distance less than d, and such that the C(2)xC(2) matrix (x(C boolean OR C')) C,C'is an element of C-2 is positive semidefinite. By the symmetry of the problem, we can apply representation theory to reduce the problem to a semidefinite programming problem with order bounded by a polynomial in n. It yields the new upper bounds A(4)(6, 3) 收起
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This paper gives spectral characterizations of two closely related graph functions: the Lovasz number v and a generalization v~1 of Delsarte's linear programming bound. There are many known characterizations of the Lovasz number v...
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This paper gives spectral characterizations of two closely related graph functions: the Lovasz number v and a generalization v~1 of Delsarte's linear programming bound. There are many known characterizations of the Lovasz number v, and each one corresponds to a similar characterization of v~1 obtained by extremizing over a larger or smaller class of objects. The spectral characterizations of v and v~1 given here involve the largest eigenvalue of a type of weighted Laplacian that Fan Chung introduced.
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We show that the Erdos-Ko-Rado inequality for t-intersecting families of k-element subsets of an n-element set can be easily extended to an inequality for cross t-intersecting families by using the eigenvalue method if n is relati...
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We show that the Erdos-Ko-Rado inequality for t-intersecting families of k-element subsets of an n-element set can be easily extended to an inequality for cross t-intersecting families by using the eigenvalue method if n is relatively large depending on k and t. The same method applies to the case of t-intersecting families of k-dimensional subspaces of an n-dimensional vector space over a finite field.
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We use the connection between positive definite functions and the character table of the symmetric group S-6 to give a short new proof of the nonexistence of a finite projective plane of order 6. For higher orders, like 10 and 12,...
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We use the connection between positive definite functions and the character table of the symmetric group S-6 to give a short new proof of the nonexistence of a finite projective plane of order 6. For higher orders, like 10 and 12, the method seems to be inconclusive as of now, but could be a basis of further research.
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摘要 :
We use the connection between positive definite functions and the character table of the symmetric group S-6 to give a short new proof of the nonexistence of a finite projective plane of order 6. For higher orders, like 10 and 12,...
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We use the connection between positive definite functions and the character table of the symmetric group S-6 to give a short new proof of the nonexistence of a finite projective plane of order 6. For higher orders, like 10 and 12, the method seems to be inconclusive as of now, but could be a basis of further research.
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We derive bounds on the size of an independent set based on eigenvalues. This generalizes a result due to Delsarte and Hoffman. We use this to obtain new bounds on the independence number of the Erdos-Renyi graphs. We investigate ...
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We derive bounds on the size of an independent set based on eigenvalues. This generalizes a result due to Delsarte and Hoffman. We use this to obtain new bounds on the independence number of the Erdos-Renyi graphs. We investigate further properties of our bounds, and show how our results on the Erdos-Renyi graphs can be extended to other polarity graphs. (c) 2007 Elsevier Inc. All rights reserved.
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For q, n, d is an element of N, let A(q)(L) (n, d) denote the maximum cardinality of a code C subset of Z(q)(n) with minimum Lee distance at least d, where Z(q) denotes the cyclic group of order q. We consider a semidefinite progr...
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For q, n, d is an element of N, let A(q)(L) (n, d) denote the maximum cardinality of a code C subset of Z(q)(n) with minimum Lee distance at least d, where Z(q) denotes the cyclic group of order q. We consider a semidefinite programming bound based on triples of codewords, which bound can be computed efficiently using symmetry reductions, resulting in several new upper bounds on A(q)(L)(n, d).
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