Journal of Combinatorial Designs最新文献

In this article, we make significant progress on a conjecture proposed by Dan Archdeacon on the existence of integer Heffter arrays $��\; ��H�\; ��\left(�\; ��m�\; �,�\; �n�\; �;�\; �s�\; �,�\; �k��\; �\right)���$ whenever the necessary conditions hold, that is, $��\; ��3�\; �⩽�\; �s�\; �⩽�\; �n��$, $��\; ��3�\; �⩽�\; �k�\; �⩽�\; �m��$, $��\; ��m�\; �s�\; �=�\; �n�\; �k��$ and $��\; ��n�\; �k�\; �\equiv �\; �0�$

We investigate the lazy burning process for Latin squares by studying their associated hypergraphs. In lazy burning, a set of vertices in a hypergraph is initially burned, and that burning spreads to neighboring vertices over time via a specified propagation rule. The lazy burning number is the minimum number of initially burned vertices that eventually burns all vertices. The hypergraphs associated with Latin squares include the $��\; ��n��$-uniform hypergraph, whose vertices and hyperedges correspond to the entries and lines (i.e., sets of rows, columns, or symbols) of the Latin square, respectively, and the 3-uniform hypergraph, which has vertices corresponding to the lines of the Latin square and hyperedges induced by its entries. Using sequences of vertices that together form a vertex cover, we show that for a Latin square of order $��\; ��n��$, the lazy burning number of its $��\; ��n��$-uniform hypergraph is bounded below by $��\; ��n^2�\; �-�\; �3�\; �n�\; �+�\; �3��$ and above by $��\; ��n^2�\; �-�\; �3�\; �n�\; �+�\; �2�\; �+�$

Terwilliger algebras are finite-dimensional semisimple algebras that were first introduced by Paul Terwilliger in 1992 in studies of association schemes and distance-regular graphs. The Terwilliger algebras of the conjugacy class association schemes of the symmetric groups $��\; ��\text{Sym}�\; ��\left(�\; �n�\; �\right)���$, for $��\; ��3�\; �\le �\; �n�\; �\le �\; �6��$, have been studied and completely determined. The case for $��\; ��\text{Sym}�\; ��\left(�\; �7�\; �\right)���$ is computationally much more difficult and has a potential application to find the size of the largest permutation codes of $��\; ��\text{Sym}�\; ��\left(�\; �7�\; �\right)���$ with a minimal distance of at least 4. In this paper, the dimension, the Wedderburn decomposition, and the block dimension decomposition of the Terwilliger algebra of the conjugacy class scheme of the group $��\; ��\text{Sym}�\; ��(�\; �$

We give some new explicit examples of putatively optimal projective spherical designs, that is, ones for which there is numerical evidence that they are of minimal size. These form continuous families, and so have little apparent symmetry in general, which requires the introduction of new techniques for their construction. New examples of interest include an 11-point spherical $��\; ���\left(�\; ��3�\; �,�\; �3��\; �\right)���$-design for $��\; ��R^3��$, and a 12-point spherical $��\; ���\left(�\; ��2�\; �,�\; �2��\; �\right)���$-design for $��\; ��R^4��$ given by four Mercedes-Benz frames that lie on equi-isoclinic planes. The latter example shows that the set of optimal spherical designs can be uncountable. We also give results of an extensive numerical study to determine the nature of the real algebraic variety of optimal projective real spherical designs, and in particular when it is a single point (a unique design) or corresponds to an infinite family of designs.