We obtain several new results contributing to the theory of real equiangular line systems. Among other things, we present a new general lower bound on the maximum number of equiangular lines in d dimensional Euclidean space; we describe the two-graphs on 12 vertices; and we investigate Seidel matrices with exactly three distinct eigenvalues. As a result, we improve on two long-standing upper bounds regarding the maximum number of equiangular lines in dimensions d=14, and d=16. Additionally, we prove the nonexistence of certain regular graphs with four eigenvalues, and correct some tables from the literature.Comment: 24 pages, to appear in JCTA. Corrected an entry in Table
Abstract. In this paper, the nonexistence of tight spherical designs is shown in some cases left open to date. Tight spherical 5-designs may exist in dimension n = (2m + 1) 2 − 2, and the existence is known only for m = 1, 2. In the paper, the existence is ruled out under a certain arithmetic condition on the integer m, satisfied by infinitely many values of m, including m = 4. Also, nonexistence is shown for m = 3. Tight spherical 7-designs may exist in dimension n = 3d 2 − 4, and the existence is known only for d = 2, 3. In the paper, the existence is ruled out under a certain arithmetic condition The concept of a spherical t-design is due to Delsarte-Goethals-Seidel [7]. For a positive integer t, a finite nonempty set X in the unit spheren−1 if the following condition is satisfied:for all polynomials f (x) = f (x 1 , x 2 , . . . , x n ) of degree not exceeding t. Here, the righthand side involves integration on the sphere, and ω n−1 denotes the volume of the sphere S n−1 . The meaning of the notion of a spherical t-design is that it is a finite set of points on the sphere that replaces the sphere itself with respect to the integration of any polynomial of degree up to t. So, it is a finite set of points on the sphere that "approximates" the sphere. It is known [7] that there is a lower bound (Fisher type inequality) for the size of a spherical t-design in S n−1 . Namely, if X is a spherical t-design in S n−1 , thenif t is even, andFrom the design-theoretical viewpoint, for t and n given, to find a t-design X with the cardinality |X| as small as possible is the most interesting problem. A 2000 Mathematics Subject Classification. Primary 05B30.
A complete classification of binary doubly even self-dual codes of length 40 is given. As a consequence, a classification of binary extremal self-dual codes of length 38 is also given. * This work was supported by JST PRESTO program.
The notion of a directed strongly regular graph was introduced by A. Duval in 1988 as one of the possible generalizations of classical strongly regular graphs to the directed case. We investigate this generalization with the aid of coherent algebras in the sense of D.G. Higman. We show that the coherent algebra of a mixed directed strongly regular graph is a non-commutative algebra of rank at least 6. With this in mind, we examine the group algebras of dihedral groups, the flag algebras of a Steiner 2-designs, in search of directed strongly regular graphs. As a result, a few new infinite series of directed strongly regular graphs are constructed. In particular, this provides a positive answer to a question of Duval on the existence of a graph with certain parameter set having 20 vertices. One more open case ୋ This paper is a revised and shortened version of the preprint [29]. This preprint stimulated a new wave of interest in the investigations of d.s.r.g.'s, in particular results by L. Jorgensen, S. A. Hobart and T.J. Shaw, A. Duval and D. Iourinski. We refer to [11,21,25,30] where a part of recent investigations is discussed with more details. We also mention that the most of the current status of the theory of d.s.r.g.'s can be found in Andries Brouwer's home page
We introduce a generalization of spin models by dropping the symmetry condition. The partition function of a generalized spin model on a connected oriented link diagram is invariant under Reidemeister moves of type II and III, giving an invariant for oriented links.
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