Abstract In structural rigidity, one studies frameworks of bars and joints in Euclidean space. Such a framework is an articulated structure consisting of rigid bars joined together at joints around which the bars may rotate. In this paper, we will describe articulated motions of realisations of hypergraphs that uses the terminology of graph of groups, and describe the motions of such a framework using group theory. Our approach allows to model a variety of situations, such as parallel redrawings, scenes, polytopes, realisations of graphs on surfaces, and even unique colourability of graphs. This approach allows a concise description of various dualities in rigidity theory. We also provide a lower bound on the dimension of the infinitesimal motions of such a framework in the special case when the underlying group is a Lie group.

Abstract Yu et al. described an algorithm for conducting computational searches for quadratic APN functions over the finite field $$\mathbb {F}_{2^n}$$ , and used this algorithm to give a classification of all quadratic APN functions with coefficients in $$\mathbb {F}_{2}$$ for dimensions n up to 9. In this paper, we speed up the running time of that algorithm by a factor of approximately $$\frac{2^n}{n^2}$$ . Based on this result, we give a complete classification of all quadratic APN functions over $$\mathbb {F}_{2^{10}}$$ with coefficients in $$\mathbb {F}_{2}$$ . We also perform some partial computations for quadratic APN functions with coefficients in $$\mathbb {F}_{2}$$ over $$\mathbb {F}_{2^{11}}$$ , and conjecture that they form 6 CCZ-inequivalent classes which also correspond to known APN functions.