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Abstract. Using an orderly algorithm, the Steiner triple systems of order 19 are classified; there are 11,084,874,829 pairwise nonisomorphic such designs. For each design, the order of its automorphism group and the number of Pasch configurations it contains are recorded; 2,591 of the designs are anti-Pasch. There are three main parts of the classification: constructing an initial set of blocks, the seeds; completing the seeds to triple systems with an algorithm for exact cover; and carrying out isomorph rejection of the final triple systems. Isomorph rejection is based on the graph canonical labeling software nauty supplemented with a vertex invariant based on Pasch configurations. The possibility of using the (strongly regular) block graphs of these designs in the isomorphism tests is utilized. The aforementioned value is in fact a lower bound on the number of pairwise nonisomorphic strongly regular graphs with parameters (57, 24,11,9).

Let F n q be a vector space of dimension n over the finite field F q . A q-analog of a Steiner system (also known as a q-Steiner system), denoted S q (t,k,n), is a set S of k-dimensional subspaces of F n q such that each t-dimensional subspace of F n q is contained in exactly one element of S. Presently, q-Steiner systems are known only for t = 1, and in the trivial cases t = k and k = n. In this paper, the first nontrivial q-Steiner systems with t 2 are constructed. Specifically, several nonisomorphic q-Steiner systems S 2 (2, 3, 13) are found by requiring that their automorphism groups contain the normalizer of a Singer subgroup of GL(13, 2). This approach leads to an instance of the exact cover problem, which turns out to have many solutions.

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LCLs or locally checkable labelling problems (e.g. maximal independent set, maximal matching, and vertex colouring) in the LOCAL model of computation are very well-understood in cycles (toroidal 1-dimensional grids): every problem has a complexity of O(1), Θ(log * n), or Θ(n), and the design of optimal algorithms can be fully automated.This work develops the complexity theory of LCL problems for toroidal 2-dimensional grids. The complexity classes are the same as in the 1-dimensional case: O(1), Θ(log * n), and Θ(n). However, given an LCL problem it is undecidable whether its complexity is Θ(log * n) or Θ(n) in 2-dimensional grids.Nevertheless, if we correctly guess that the complexity of a problem is Θ(log * n), we can completely automate the design of optimal algorithms. For any problem we can find an algorithm that is of a normal form A • S k , where A is a finite function, S k is an algorithm for finding a maximal independent set in kth power of the grid, and k is a constant.Finally, partially with the help of automated design tools, we classify the complexity of several concrete LCL problems related to colourings and orientations. arXiv:1702.05456v2 [cs.DC] 24 May 2017 1.1 Problem setting: LCL problems on grids 92 33 77 57 49 26 71 79 8 62 48 24 31 21 15 30 60 67 0 5 17 95 23 47 87 80 25 38 20 64 45 61 91 51 69 1 74 55 3 98 88 99 58 53 63 40 16 2 39Grids. In this work, we study distributed algorithms in a setting where the underlying input graph is a grid. Specifically, we consider the complexity of locally checkable labelling problems, or LCL problems, in the standard LOCAL model of distributed complexity, and consider graphs that are toroidal two-dimensional n × n grids with a consistent orientation; we focus on the two-dimensional case for concreteness, but most of our results generalise to d-dimensional grids of arbitrary dimensions. This setting occupies a middle ground between the wellunderstood directed n-cycles [10,32], where all solvable LCL problems are known to have deterministic time complexity either O(1), Θ(log * n) or Θ(n), and the more complicated setting of general n-vertex graphs, where intermediate problems with time complexities such as Θ(log n) are known to exist, even for bounded-degree graphs. Grid-like systems with local dynamics also occur frequently in the study of real-world phenomena. However, grids have so far not been systematically studied from a distributed computing perspective.LOCAL model and LCL problems. In the LOCAL model of distributed computing, nodes are labelled with unique numerical identifiers with O(log n) bits. A time-t algorithm in this model is simply a mapping from radius-t neighbourhoods to local outputs; equivalently, it can be interpreted as a message-passing algorithm in which the nodes exchange messages for t synchronous rounds and then announce their local outputs.LCL problems are graph problems for which the feasibility of a solution can be verified by checking the solution for each O(1)-radius neighbourhood; if all local neighbourhoods look valid, the s...