  We generalise the existence of combinatorial designs to the setting of subset
sums in lattices with coordinates indexed by labelled faces of simplicial
complexes. This general framework includes the problem of decomposing
hypergraphs with extra edge data, such as colours and orders, and so
incorporates a wide range of variations on the basic design problem, notably
Baranyai-type generalisations, such as resolvable hypergraph designs, large
sets of hypergraph designs and decompositions of designs by designs. Our method
also gives approximate counting results, which is new for many structures whose
existence was previously known, such as high dimensional permutations or Sudoku
squares.
