Robert F. Bailey

Error-correcting codes from permutation groups

We replace the usual setting for error-correcting codes (i.e. vector spaces over finite fields) with that of permutation groups. We give an algorithm which uses a combinatorial structure which we call an uncovering-by-bases, related to covering designs, and construct some examples of these. We also analyse the complexity of the algorithm. We then formulate a conjecture about uncoverings-by-bases, for which we give some supporting evidence and prove for some special cases. In particular, we consider the case of the symmetric group in its action on 2-subsets, where we make use of the theory of graph decompositions. Finally, we discuss the implications this conjecture has for the complexity of the decoding algorithm.

Hamiltonian decompositions of complete k-uniform hypergraphs

Using a generalisation of Hamiltonian cycles to uniform hypergraphs due to Katona and Kierstead, we define a new notion of a Hamiltonian decomposition of a uniform hypergraph. We then consider the problem of constructing such decompositions for complete uniform hypergraphs, and describe its relationship with other topics, such as design theory.

Uncoverings-by-bases for base-transitive permutation groups

AbstractAn uncovering-by-bases for a group G acting on a finite set Ω is a set

Error-correcting codes from k-resolving sets

\mathcal{U}

On the single-orbit conjecture for uncoverings-by-bases

of bases for G such that any r-subset of Ω is disjoint from at least one base in

On the metric dimension of Grassmann graphs

\mathcal{U}