The -optimal and minimax-optimal supersaturated designs (SSDs) with 12 rows, columns, and are enumerated in a computer search: there are, respectively, 34, 146, 0, 3, and 1 such designs for , and 6. Cheng and Tang proved that for , there are no such SSDs. This completes the enumeration of all SSDs with the described restrictions. These results are obtained by enumerating the
We obtain, by computer, a full classification up to equivalence of all minimal blocking sets with a nontrivial automorphism group in the Desarguesian projective plane of order 11. We list the resulting numbers of sets according to their size and the order of their automorphism group. For the minimal blocking sets with the larger automorphism groups, explicit descriptions are given. We also give a list of all blocking semiovals among the results. In contrast to similar work on the planes of smaller order, only those blocking sets were generated that have an automorphism group of size , as the number of minimal blocking sets with a trivial group is estimated to be infeasibly large. New algorithms had to be devised to obtain these results because simply generating all sets and filtering out those with a nontrivial automorphism group was totally impractical.
We computationally completely enumerate a number of types of row-column designs up to isotopism, including double, sesqui, and triple arrays as known from the literature, and two newly introduced types that we call mono arrays and AO-arrays. We calculate autotopism group sizes for the designs we generate. For larger parameter values, where complete enumeration is not feasible, we generate examples of some of the designs, and for some admissible parameter sets, we prove non-existence results. We give some explicit constructions of sesqui arrays, mono arrays and AO-arrays, in particular, we prove constructively that AO-arrays exist for all feasible parameter sets. Finally, we investigate connections to Youden rectangles and binary pseudo-Youden designs.
One method of constructing -SEDFs (i.e., strong external difference families) in makes use of -valuations of complete bipartite graphs . We explore this approach and we provide a classification theorem which shows that all such -valuations can be constructed recursively via a sequence of “blow-up” operations. We also enumerate all
A partial partition of an -element set is a collection of pairwise disjoint -element subsets. It is proved that, if is large enough, one can find such partial partitions in such a way that if and
In this article, we make significant progress on a conjecture proposed by Dan Archdeacon on the existence of integer Heffter arrays whenever the necessary conditions hold, that is, , , and
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 -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 , the lazy burning number of its -uniform hypergraph is bounded below by and above by
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