Journal of Combinatorial Designs最新文献

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 $��\; ��r��$.

We generalize the well-known Oberwolfach problem posed by Ringel in 1967. We suppose to have $��\frac{v}{2}�$ couples (here $��v�\; �\ge �\; �4�$ is an even integer) and suppose that they have to be seated for several nights at $��t�$ round tables in such a way that each person seats next to his partner exactly $��r�\; �\ge �\; �0�$ times and next to every other person exactly once. We call this problem the Oberwolfach problem with loving couples. When $��r�\; �=�\; �0�$, the problem coincides with the so-called spouse-avoiding variant, which was introduced by Huang, Kotzig, and Rosa in 1979. While if either $��r�\; �=�\; �2�$ or $��r�$ equals the number of nights, it corresponds to the spouse-loving variant or to the Honeymoon variant, which was recently studied by Bolohan et al. and by Lepine and Sajna, respectively. In this paper, for each possible choice of $��r�$, we construct many classes of solutions to the Oberwolfach problem with loving couples. We also obtain new solutions to the Honeymoon variant.

A multifold 1-perfect code (1-perfect code for list decoding) in any graph is a set $�\; ��\; �C��$ of vertices such that every vertex of the graph is at distance not more than 1 from exactly $�\; ��\; �\mu ��$ elements of $�\; ��\; �C��$. In $�\; ��\; �q��$-ary Hamming graphs, where $�\; ��\; �q��$ is a prime power, we characterize all parameters of multifold 1-perfect codes and all parameters of additive multifold 1-perfect codes. In particular, we show that additive multifold 1-perfect codes are related to special multiset generalizations of spreads, multispreads, and that multispreads of parameters corresponding to multifold 1-perfect codes always exist.

This paper is a contribution to the classification of all pairs $��\left(��T�\; �,�\; �G��\; �\right)�$, where $��T�$ is a triple system and $��G�$ is a block-transitive but not flag-transitive automorphism group of $��T�$. We prove that if the socle of $��G�$ is a sporadic or alternating group, then one of the following holds: