Journal of Combinatorial Designs最新文献

In this paper, we characterise point sets having the same intersection numbers with respect to hyperplanes as an ovoidal cone. In particular, we show that a set of points of $��\text{PG}��\left(��4�\; �,�\; �q��\; �\right)��$ which blocks all planes and intersects solids in $��q�\; �+�\; �1�$, $��q^2�\; �+�\; �1�$ or $��q^2�\; �+�\; �q�\; �+�\; �1�$ points is a plane or an ovoidal cone, and determine all examples that arise when the blocking condition is omitted.

An internal or friendly partition of a vertex set $��\; ��V�\; ��\left(�\; �G�\; �\right)���$ of a graph $��\; ��G��$ is a partition to two nonempty sets $��\; ��A�\; �\cup �\; �B��$ such that every vertex has at least as many neighbours in its own class as in the other one. Motivated by Diwan's existence proof on internal partitions of graphs with high girth, we give constructive proofs for the existence of internal partitions in the incidence graph of projective planes and discuss its geometric properties. In addition, we determine exactly the maximum possible difference between the sizes of the neighbour set in its own class and the neighbour set of the other class that can be attained for all vertices at the same time for the incidence graphs of Desarguesian planes of square order.

In a nesting of a balanced incomplete block design (or BIBD), we wish to add a point (the nested point) to every block of a $��\; ���\left(�\; ��v�\; �,�\; �k�\; �,�\; �\lambda ��\; �\right)���$-BIBD in such a way that we end up with a partial $��\; ���\left(�\; ��v�\; �,�\; �k�\; �+�\; �1�\; �,�\; �\lambda �\; �+�\; �1��\; �\right)���$-BIBD. In the case where the partial $��\; ���(�\; ��v�\; �,�\; �k�\; �+�\; �1�\; �,�\; �\lambda �\; �$

We introduce and explore a new concept of evasive subspace with respect to a collection of subspaces sharing a common dimension, most notably partial spreads. We show that this concept generalises known notions of subspace scatteredness and evasiveness. We establish various upper bounds for the dimension of an evasive subspace with respect to arbitrary partial spreads, obtaining improvements for the Desarguesian ones. We also establish existence results for evasive spaces in a nonconstructive way, using a graph theory approach. The upper and lower bounds we derive have a precise interpretation as bounds for the critical exponent of certain combinatorial geometries. Finally, we investigate connections between the notion of evasive space we introduce and the theory of rank-metric codes, obtaining new results on the covering radius and on the existence of minimal vector rank-metric codes.

This paper lies in the context of the studies of eigenfunctions of graphs having minimum cardinality of support. One of the tools is the weight-distribution bound, a lower bound on the cardinality of support of an eigenfunction of a distance-regular graph corresponding to a nonprincipal eigenvalue. The tightness of the weight-distribution bound was previously shown in general for the smallest eigenvalue of a Grassmann graph. However, a characterisation of optimal eigenfunctions was not obtained. Motivated by this open problem, we consider the class of strongly regular Grassmann graphs and give the required characterisation in this case. We then show the tightness of the weight-distribution bound for block graphs of affine designs (defined on the lines of an affine space with two lines being adjacent when intersect) and obtain a similar characterisation of optimal eigenfunctions.

We propose to consider a mutual incidence matrix $��\; ��M��$ of two balanced incomplete block designs built on the same finite set. In the simplest case, this matrix reduces to the standard incidence matrix of one block design. We find all eigenvalues of the matrices $��\; ��M�\; �M^T��$ and $��\; ��M^T�\; �M��$ and their eigenspaces.

In this paper, we show an infinite series of 3-designs in the extended quadratic residue codes over $��\; ��F_{r^2}��$ for a prime $��\; ��r��$.