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Let S n denote the symmetric group on [n] = {1,. .. , n}. A family I ⊆ S n is intersecting if any two elements of I have at least one common entry. It is known that the only intersecting families of maximal size in S n are the cosets of point stabilizers. We show that, under mild restrictions, analogous results hold for the alternating group and the direct… (More)

Quantum synchronizable codes are quantum error-correcting codes that can correct the effects of quantum noise as well as block synchronization errors. We improve the previously known general framework for designing quantum synchronizable codes through more extensive use of the theory of finite fields. This makes it possible to widen the range of tolerable… (More)

We consider integer matrices N t (h) whose rows are indexed by the t-subsets of an n-set and whose columns are all images of a particular column h under the symmetric group S n. Earlier work has determined a diagonal form for N t (h) when h has at least t 'isolated vertices' and the results were applied to the binary case of a zero-sum Ramsey-type problem… (More)

Let G be a subgraph of a complete bipartite graph K n,n. Let N (G) be a 0-1 incidence matrix with edges of K n,n against images of G under the automorphism group of K n,n. A diagonal form of N (G) is found for every G, and whether the row space of N (G) over Z p contains the vector of all 1's is determined. This reproves Caro and Yuster's results on… (More)

- Wing Hong, Tony Wong, Richard M. Wilson
- 2013

All Rights Reserved iii To my wife, Jane, who offers me unconditional love, care and support throughout the course of this thesis. iv Acknowledgements First and foremost, I would like to express my deepest gratitude to my advisor, Prof. Richard M. Wilson, for his guidance, teaching, patience and support. His extensive knowledge in combinatorics and… (More)

We consider integer matrices N t whose rows are indexed by the t-subsets of an n-set and whose columns are all distinct images of a particular column under the symmetric group S n. Examples include matrices in the association algebras of the Johnson schemes. Three related problems are addressed. What is the Smith normal form (or a diagonal form) for N t and… (More)

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