Strong external difference families (SEDFs) have applications to cryptography and are rich combinatorial structures in their own right; until now, all SEDFs have been in abelian groups. In this paper, we consider SEDFs in both abelian and non-abelian groups. We characterize the order of groups possessing admissible parameters for non-trivial SEDFs, develop non-existence and existence results, several of which extend known results, and present the first family of non-abelian SEDFs. We introduce the concept of equivalence for EDFs and SEDFs, and begin the task of enumerating SEDFs. Complete results are presented for all groups up to order 24, underpinned by a computational approach.
Pen and paper puzzles like Sudoku, Futoshiki and Skyscrapers are hugely popular. Solving such puzzles can be a trivial task for modern AI systems. However, most AI systems solve problems using a form of backtracking, while people try to avoid backtracking as much as possible. This means that existing AI systems do not output explanations about their reasoning that are meaningful to people. We present DEMYSTIFY, a tool which allows puzzles to be expressed in a high-level constraint programming language and uses MUSes to allow us to produce descriptions of steps in the puzzle solving. We give several improvements to the existing techniques for solving puzzles with MUSes, which allow us to solve a range of significantly more complex puzzles and give higher quality explanations. We demonstrate the effectiveness and generality of DEMYSTIFY by comparing its results to documented strategies for solving a range of pen and paper puzzles by hand, showing that our technique can find many of the same explanations.
The normaliser problem takes as input subgroups G and H of the symmetric group S n , and asks one to compute N G (H). The fastest known algorithm for this problem is simply exponential, whilst more efficient algorithms are known for restricted classes of groups. In this paper, we will focus on groups with many orbits. We give a new algorithm for the normaliser problem for these groups that performs many orders of magnitude faster than previous implementations in GAP. We also prove that the normaliser problem for the special case G = S n is at least as hard as computing the group of monomial automorphisms of a linear code over any field of fixed prime order.
No abstract
We introduce a new framework for solving an important class of computational problems involving finite permutation groups, which includes calculating set stabilisers, intersections of subgroups, and isomorphisms of combinatorial structures. Our techniques are inspired by and generalise 'partition backtrack', which is the current state-of-the-art algorithm introduced by Jeffrey Leon in 1991. But, instead of ordered partitions, we use labelled directed graphs to organise our backtrack search algorithms, which allows for a richer representation of many problems while often resulting in smaller search spaces. In this article we present the theory underpinning our framework, we describe our algorithms, and we show the results of some experiments. An implementation of our algorithms is available as free software in the GraphBacktracking package for GAP.
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