Linear spaces with a line-transitive point-imprimitive automorphism group and Fang-Li parameter gcd(k,r) at most eight

In 1991, Weidong Fang and Huiling Li proved that there are only finitely many non-trivial linear spaces that admit a line-transitive, point-imprimitive group action, for a given value of gcd(k,r), where k is the line size and r is the number of lines on a point. The aim of this paper is to make that result effective. We obtain a classification of all linear spaces with this property having gcd(k,r) at most 8. To achieve this we collect together existing theory, and prove additional theoretical restrictions of both a combinatorial and group theoretic nature. These are organised into a series of algorithms that, for gcd(k,r) up to a given maximum value, return a list of candidate parameter values and candidate groups. We examine in detail each of the possibilities returned by these algorithms for gcd(k,r) at most 8, and complete the classification in this case.Comment: 47 pages Version 1 had bbl file omitted. Apologie

Classification of flocks of the quadratic cone over fields of order at most 29

Peer Reviewe

Tight sets and m-ovoids of finite polar spaces

An intriguing set of points of a generalised quadrangle was introduced in [2] as a unification of the pre-existing notions of tight set and m-ovoid. It was shown in [2] that every intriguing set of points in a finite generalised quadrangle is a tight set or an m-ovoid (for some m). Moreover, it was shown that an m-ovoid and an i-tight set of a common generalised quadrangle intersect in mi points. These results yielded new proofs of old results, and in this paper, we study the natural analogue of intriguing sets in finite polar spaces of higher rank. In particular, we use the techniques developed in this paper to give an alternative proof of a result of Thas [36] that there are no ovoids of H(2r, q2), Q−(2r+1, q), andW(2r−1, q) for r> 2. We also strengthen a result of Drudge on the non-existence of tight sets in W(2r−1, q), H(2r + 1, q2), and Q+(2r + 1, q), and we give a new proof of a result of due to De Winter, Luyckx, and Thas [9,28] that an m-system ofW(4m+3, q) or Q−(4m+3, q) is a pseudo-ovoid of the ambient projective space

GAP 4 Package FinInG:Finite Incidence Geometry

FinInG is a package for computation in finite geometry. It provides users with the basic tools to work in various areas of finite geometry from the realms of projective spaces to the flat lands of generalised polygons. The algebraic power of GAP is employed, particularly in its facility with matrix and permutation groups