Symmetric conference matrices of order pq2+1

Introduction and definitions. A conference matrix of order n is a square matrix C with zeros on the diagonal and zbl elsewhere, which satisfies the orthogonality condition CC T = (n -1)1. If in addition C is symmetric, C

Perp-systems and partial geometries

A perp-system R(r) is a maximal set of r-dimensional subspaces of PG(N,q) equipped with a polarity rho, such that the tangent space of an element of R(r) does not intersect any element of R(r). We prove that a perp-system yields partial geometries, strongly regular graphs, two-weight codes, maximal arcs and k-ovoids. We also give some examples, one of them yielding a new pg(8,20,2)

On the Uniform Random Generation of Non Deterministic Automata Up to Isomorphism

In this paper we address the problem of the uniform random generation of non deterministic automata (NFA) up to isomorphism. First, we show how to use a Monte-Carlo approach to uniformly sample a NFA. Secondly, we show how to use the Metropolis-Hastings Algorithm to uniformly generate NFAs up to isomorphism. Using labeling techniques, we show that in practice it is possible to move into the modified Markov Chain efficiently, allowing the random generation of NFAs up to isomorphism with dozens of states. This general approach is also applied to several interesting subclasses of NFAs (up to isomorphism), such as NFAs having a unique initial states and a bounded output degree. Finally, we prove that for these interesting subclasses of NFAs, moving into the Metropolis Markov chain can be done in polynomial time. Promising experimental results constitute a practical contribution.Comment: Frank Drewes. CIAA 2015, Aug 2015, Umea, Sweden. Springer, 9223, pp.12, 2015, Implementation and Application of Automata - 20th International Conferenc

Mixed partitions of sets of triples into small planes

We continue our study of partitions of the set of all ((v)(3)) triples chosen from a v-set into pairwise disjoint planes with three points per line. We develop further necessary conditions for the existence of partitions of such sets into copies of PG(2, 2) and copies of AG(2, 3), and deal with the cases v = 13, 14, 15 and 17. These partitions, together with those already known for v = 12, 16 and 18, then become starters for recursive constructions of further infinite families of partitions. (C) 2004 Elsevier B.V. All rights reserved

Strongly regular (alpha,beta)-geometries

In this paper we introduce strongly regular (alpha, beta)-geometries. These are a class of geometries that generalise semipartial geometries. Like semipartial geometries the underlying point graph is strongly regular and this is part of the motivation for studying the geometries. In the paper several necessary conditions for existence are given. Strongly regular (alpha, beta)-reguli are defined, and it is shown how they may be used to construct strongly regular (alpha, beta)-geometries. This generalises similar results by J. A. Thas in (1980, European J. Combin. 1, 189-192) constructing semipartial geometries. Several constructions of strongly regular (alpha, beta)-geometries are given, and possible parameters of existence for small cases are listed. (C) 2001 Academic Press

Partitions of sets of designs on seven, eight and nine points

We consider the sets of all possible Steiner triple systems (STS) which can be defined on a 7-set or an 8-set, the sets of all possible Steiner quadruple systems (SQS) which can be defined on an 8-set or a 9-set, and the set of all possible Steiner triple systems, on 9 points each, which can be defined on a 9-set. By considering the large and overlarge sets of these designs, we derive various strongly regular graphs and balanced or partially balanced designs. We show connections between the overlarge sets of SQS(8), the STS(9) and the resolutions of the set of all () triples chosen from a 9-set into 28 parallel classes of three pairwise disjoint triples, with no two parallel classes orthogonal. Finally we show that the set of all 840 distinct STS(9)'s which can be defined on a given 9-set can be partitioned into 120 large sets of STS(9)

Partitioning sets of triples into small planes

We study partitions of the set of all ((v)(3)) triples chosen from a v-set into pairwise disjoint planes with three points per line. Our partitions may contain copies of PG(2, 2) only (Fano partitions) or copies of AG(2, 3) only (affine partitions) or copies of some planes of each type (mixed partitions). We find necessary conditions for Fano or affine partitions to exist. Such partitions are already known in several cases: Fano partitions for v = 8 and affine partitions for v = 9 or 10. We construct such partitions for several sporadic orders, namely, Fano partitions for v = 14, 16, 22, 23, 28, and an affine partition for v = 18. Using these as starter partitions, we prove that Fano partitions exist for v = 7(n) + 1, 13(n) + 1, 27(n) + 1, and affine partitions for v = 8(n) + 1, 9(n) + 1, 17(n) + 1. In particular, both Fano and affine partitions exist for v = 3(6n) + 1. Using properties of 3-wise balanced designs, we extend these results to show that affine partitions also exist for v = 3(2n). Similarly, mixed partitions are shown to exist for v = 8(n), 9(n), 11(n) + 1

Overlarge sets of 2-(11, 5, 2) designs and related configurations

We consider the construction of several configurations, including: • overlarge sets of 2-(11,5,2) designs, that is, partitions of the set of all 5-subsets of a 12-set into 72 2-(11,5,2) designs; • an indecomposable doubly overlarge set of 2-(11,5,2) designs, that is, a partition of two copies of the set of all 5-subsets of a 12-set into 144 2-(11,5,2) designs, such that the 144 designs can be arranged into a 12 × 12 square with interesting row and column properties; • a partition of the Steiner system S(5,6,12) into 12 disjoint 2-(11,6,3) designs arising from the diagonal of the square; • bidistant permutation arrays and generalized Room squares arising from the doubly overlarge set, and their relation to some new strongly regular graphs