Journal of Combinatorial Designs最新文献

Maximum rank-distance (MRD) codes are (not necessarily linear) maximum codes in the rank-distance metric space on $��m�$-by-$��n�$ matrices over a finite field $��F_q�$. They are diameter perfect and have the cardinality $��q^{�m��\left(��n�-�d�+�1��\right)��}�$ if $��m�\ge �n�$. We define switching in MRD codes as the replacement of special MRD subcodes by other subcodes with the same parameters. We consider constructions of MRD codes admitting switching, such as punctured twisted Gabidulin codes and direct-product codes. Using switching, we construct a huge class of MRD codes whose cardinality grows doubly exponentially in $��m�$ if the other parameters ($��n�,��q�$, the code distance) are fixed. Moreover, we construct MRD codes with different affine ranks and aperiodic MRD codes.

Recently, new combinatorial structures called disjoint partial difference families (DPDFs) and external partial difference families (EPDFs) were introduced, which simultaneously generalize partial difference sets, disjoint difference families and external difference families, and have applications in information security. So far, all known construction methods have used cyclotomy in finite fields. We present the first noncyclotomic infinite families of DPDFs which are also EPDFs, in structures other than finite fields (in particular cyclic groups and nonabelian groups). As well as direct constructions, we present an approach to constructing DPDFs/EPDFs using relative difference sets (RDSs); as part of this, we demonstrate how the well-known RDS result of Bose extends to a very natural construction for DPDFs and EPDFs.

We describe a class of combinatorial design problems which typically occur in professional sailing league competitions. We discuss connections to resolvable block designs and equitable coverings and to scheduling problems in operations research. We in particular give suitable boolean quadratic and integer linear optimization problem formulations, as well as further heuristics and restrictions, that can be used to solve sailing league problems in practice. We apply those techniques to three case studies obtained from real sailing leagues and compare the results with previously used tournament plans.

In this article, we study symmetric designs admitting flag-transitive, point-imprimitive almost simple automorphism groups with socle sporadic simple groups. As a corollary, we present a classification of symmetric designs admitting flag-transitive automorphism group whose socle is a sporadic simple group, and in conclusion, there are exactly seven such designs, one of which admits a point-imprimitive automorphism group and the remaining are point-primitive.

It is shown that the flag-transitive, point-primitive automorphism groups of 2-$��\left(�\; ��v�\; �,�\; �k�\; �,�\; �\lambda ��\; �\right)�$ designs with $��\lambda �$ prime must be of affine type or almost simple type.

A set of edges $��\Gamma �$ of a graph $��G�$ is an edge dominating set if every edge of $��G�$ intersects at least one edge of $��\Gamma �$, and the edge domination number $��{\gamma }_e�\; ��\left(�\; �G�\; �\right)��$ is the smallest size of an edge dominating set. Expanding on work of Laskar and Wallis, we study $��{\gamma }_e�\; ��\left(�\; �G�\; �\right)��$ for graphs $��G�$ which are the incidence graph of some incidence structure $��D�$, with an emphasis on the case when $��D�$ is a symmetric design. In particular, we show in this latter case that determining $�$

Let $�\; ��D�\; �=�\; ��\left(�\; ��P�\; �,�\; �ℬ��\; �\right)��$ be a nontrivial symmetric $�\; ���\left(�\; ��v�\; �,�\; �k�\; �,�\; �\lambda ��\; �\right)��$-design with $�\; ��\lambda �\; �\le �\; �100�$, and let $�\; ��G�$ be a flag-transitive automorphism group of $�\; ��D�$. In this paper, we show that if $�\; ��G�$ is quasiprimitive on $�\; ��P�$, then $�\; ��G�$ is of holomorph affine or almost simple type. Moreover, if $�\; ��G�$ is imprimitive on $�\; ��P�$, then $�\; ��G�$ is of almost simple type. According to this observation and to the classification of the finite simple groups we determine all such symmetric designs and the corresponding automorphism groups. We conclude with two open problems and a conjecture.

In this paper, we give a solution to the last outstanding case of the directed Oberwolfach problem with tables of uniform length. Namely, we address the two-table case with tables of equal odd length. We prove that the complete symmetric digraph on $��2�m�$ vertices, denoted $��K_{�2�m�}^{*}�$, admits a resolvable decomposition into directed cycles of odd length $��m�$. This completely settles the directed Oberwolfach problem with tables of uniform length.