Publications
I started off working algebraic design theory, specifically with group actions on Hadamard matrices and related combinatorial designs. I've looked at applications of linear algebra and combinatorics to problems in compressed sensing, and private communication. Recently I've been interested in gathering together the techniques of algebraic design theory, with the goal of pushing them in new directions - towards quantum information theory, perhaps.
1. On Twin Prime Power Hadamard matrices, 2010, Cryptography and Communications - Discrete Structures, Boolean Functions and Sequences, 2, 2, pp. 261-269.
Joint work with Richard Stafford. In this paper we showed that the only Hadamard matrix constructed from a twin prime power difference set which is cocyclic has order 16, answering a question of Kathy Horadam.
2. The cocyclic Hadamard matrices of order less than 40, 2011, Designs, Codes and Cryptography, 58, 1, pp. 73-88.
Joint work with Marc Roder. This paper exploits a known relation between certain relative difference sets and cocyclic Hadamard matrices to obtain some new classification results.
3. Difference sets and doubly transitive actions on Hadamard matrices, 2012, Journal of Combinatorial Theory, Series A, 119, 6, pp. 1235-1249.
This paper extends the results of [1] to a classification of all cocyclic Hadamard matrices which can be constructed from difference sets. As part of the analysis, the paper contains a description of a new family of skew-Hadamard difference sets related to Paley type I matrices.
4. Inequivalence of difference sets: On a remark of Baumert, 2013, Electronic Journal of Combinatorics, 20, 1, P.38.
This paper provides a complete and self contained proof of an often cited remark contained in Baumert's "Cyclic difference sets".
5. Nesting symmetric designs, 2013, Bulletin of the Irish Mathematical Society, 72, pp. 71-74.
A short note classifying the symmetric designs which can be 'nested', answering a question posed by Darryn Bryant.
6. An asymptotic existence result on compressed sensing matrices, 2015, Linear Algebra and its Applications, 475, 134--150.
Joint work with Darryn Bryant. We introduce a generalisation of equiangular tight frames, adapt a construction of Seidel and generalise a result of Fickus, Mixon and Tremain. We prove some new results on the existence of pairwise balanced designs with specified numbers of blocks, and obtain new existence results on compressed sensing matrices as a corollary.
7. Trades in complex Hadamard matrices, 213--221, 2015, Algebraic design theory with Hadamard matrices, Springer Proceedings in Mathematics and Statistics.
Joint work with Ian Wanless. We generalise a result of Alon on the number sparse linear combinations of the rows of a complex Hadamard matrix. We bound the size of many trades in complex Hadamard matrices, and relate these results to previous work on small rank submatrices of Hadamard matrices.
8. Classifying cocyclic Butson Hadamard matrices, 93--106, 2015, Algebraic design theory with Hadamard matrices, Springer Proceedings in Mathematics and Statistics.
Joint work with Ronan Egan and Dane Flannery. We develop two computational frameworks for the classification of complex Hadamard matrices, and apply them to certain small parameter sets. We also describe an algorithm for testing equivalence of Butson Hadamard matrices.
9. Explicit correlation amplifiers for finding outlier correlations in deterministic sub quadratic time, 24th European Symposium on Algorithms (ESA) 2016.
Joint work with Petteri Kaski, Matti Karppa and Jukka Kohonen. We derandomise an algorithm of G. Valiant for finding correlated pairs of binary vectors subject to some mild assumptions.
10. Compressed sensing with combinatorial designs: theory and simulations, IEEE Transaction on Information Theory, 63, 8, 4850--4859, 2017.
Joint work with Darryn Bryant, Charles Colbourn and Daniel Horsley. We continue the analysis of the construction of compressed sensing matrices presented in [6], using both deterministic and probabilistic methods. We also provide extensive simulations.
11. Improved User-Private Information Retrieval with Finite Geometry, 10th International Workshop on Coding and Cryptography (WCC) 2017.
Joint work with Oliver Gnilke, Marcus Greferath, Camilla Hollanti, Guillermo Nunez Ponasso and Eric Swartz. We generalize work of Swanson and Stinson to construct communication protocols in which most users retain privacy even after the defection of a coalition of users whose size grows as a root function of the total number of users. Previous protocols could be compromised by a finite number of users.
12. Remark on a result of Constantine, Annals of Combinatorics, 22, 1, 93--97, 2018.
This short note gives a construction of some self-complementary binary codes from Hadamard matrices, improving on a result of Constantine. We also show that these codes are close to the Grey-Rankin bound.
13. Sparsification of Matrices and Compressed Sensing, Bulletin of the Irish Mathematical Society, 81, pp. 5--22, 2018.
Joint work with Fintan Hegarty and Yun-Bin Zhao. In this largely computational paper, we explore the effect of sparsification on compressed sensing performance for a range of matrix constructions and recovery algorithms.
14. Morphisms of Butson matrices, Linear Algebra and its Applications, 577, pp. 78--93, 2019.
Joint work with Ronan Egan. We develop necessary and sufficient conditions for the existence of generalised Kronecker-type products of Hadamard matrices in which the product matrix is expressed over a smaller field than the input matrices. This generalizes constructions of Turyn and Compton-Craigen-de Launey.
15. Construction of the outer automorphism of S6 via a complex Hadamard matrix, Mathematics in Computer Science, 12, 4, 453--458, 2018.
Joint work with Neil Gillespie and Cheryl Praeger. We construct a representation of a non-split extension of the symmetric group on six points over the split-quaternions. The outer automorphism of S6 can then be realized as conjugation by a certain complex Hadamard matrix.
16. Spectra of Hadamard matrices, Australasian Journal of Combinatorics, 73, 3, 501--512, 2019.
Joint work with Ronan Egan and Eric Swartz. We construct Hadamard matrices with specified characteristic polynomials. These can be used to give new examples of complete morphisms of Butson matrices.
17. Improved user-private information retrieval via finite geometry, Designs, Codes and Cryptography, 87, 2-3, 665--677, 2019.
Joint work with Oliver Gnilke, Marcus Greferath, Camilla Hollanti, Guillermo Nunez Ponasso and Eric Swartz. Journal version of publication 11 on this list. We extended that paper to include analysis of the known families of generalised quadrangles.
18. Constructing cocyclic Hadamard matrices of order 4p, Journal of Combinatorial Designs, 27, 11, 627--642, 2019.
Joint work with Santiago Barrera-Acevedo and Heiko Dietrich. De Launey and Flannery produced a taxonomy of cocyclic Hadamard matrices of order 4p where p is prime. We prove that many cases produce equivalent Hadamard matrices. If p = 3 mod 4, every cocyclic Hadamard matrix is equivalent either to a Williamson-type matrix or an Ito-type matrix, if p = 1 mod 4, there are three additional possibilities. We use these results to classify the cocyclic Hadamard matrices of orders 44 and 52.
19. Homomorphisms of matrix algebras and constructions of Butson-Hadamard matrices, Discrete Mathematics, 342, 12, 111606, 2019.
Joint work with Eric Swartz. We generalise a result of Ostergard and Paavola on the existence of morphisms between certain classes of Butson-Hadamard matrices. Our proof is algebraic, short and straightforward.
20. Dimensions of semisimple matrix algebras, Pi Mu Epsilon Journal, 15, 1, 31--38, 2019.
Joint work with Phillip Heikoop. In this paper, aimed at advanced undergraduates, we sketch the structure theory of associative algebras. We adapt an argument of Savitt and Stanley to show that every dimension in the range [0, n^2 - (9/4) n^(3/2) ] is the dimension of a semisimple subalgebra of the matrix algebra of order n. This paper grew from Phillip's final year thesis.
21. Explicit correlation amplifiers for finding outlier correlations in deterministic sub quadratic time, Algorithmica, 82, 11, 3306--3337, 2020.
Joint work with Petteri Kaski, Matti Karppa and Jukka Kohonen. Journal version of the work that appeared in conference proceedings at (9) above.
22. Morphisms of skew Hadamard matrices, Bulletin of the Institute of Combinatorics and its Applications, 90, 50--62, 2020.
Joint work with Phillip Heikoop, John Pugmire and Guillermo Nunez Ponasso. In this paper, we construct a morphism from the Quaternary Unit Hadamard matrices of Fender, Kharaghani and Suda to real Hadamard matrices. A special case of this work recovers a result of Mukhopadhyay and Seberry.
23. Cocyclic two-circulant core Hadamard matrices, Journal of Algebraic Combinatorics, 55, 201--215, 2022.
Joint work with Santiago Barrera-Acevedo and Heiko Dietrich. A question of Horadam asks whether Hadamard matrices of order 2k+2 constructed using using two circulant cores of order k are cocyclic. We identify the possible orders for Hadamard matrices of this form, and resolve the existence question for orders up to 511.
24. A survey of the Hadamard maximal determinant problem, Electronic Journal of Combinatorics, P4.41, 2021.
Joint work with Patrick Browne, Ronan Egan and Fintan Hegarty. We give a historically informed survey of work on the maximal determinant problem, with particular focus on orders not divisible by 4. Some of these results appear in English for the first time.
25. Good sequencings of partial Steiner systems, Designs, Codes and Cryptography, 90, 2375--2383, 2022.
Joint work with Daniel Horsley. A sequencing of a Steiner system is a cyclic ordering of the points. The sequencing is r-good if no block is contained in a set of r consecutive points. Using probabilistic methods and bounds on the indepdence numbers of Steiner systems due to Kostochka, Mubayi and Verstraete, we prove the existence of Steiner triple systems which are asymptotically optimally good.
26. Lights out on the Hypercube over the ring of integers modulo k, Pi Mu Epsilon Journal, 15, 7, 415--421, 2022.
Joint work Travis Peters and James Schwinghamer. A short note for undergraduates in which we relate the eigenvalues and eigenspaces of cube graphs to solvability of the lights out game.
27. Segre's theorem on ovals in Desarguesian projective planes, Bulletin of the Irish Mathematical Society, 91, 37--48, 2023.
Joint work with Patrick Browne and Steven Dougherty. An exposition of Segre's theorem which shows that an oval in a projective plane of odd order is in fact a conic. While mostly expository, we make some simplifications to Segre's proof.
28. Invariants of Quadratic forms and applications in Design theory, Linear Algebra and its Applications, 682, 1--27, 2024.
Joint work with Oliver Gnilke, Oktay Olmez and Guillermo Nunez Ponasso. In this survey paper, we construct invariants of rational quadratic forms, and use them to give a complete proof of the Bruck-Ryer-Chowla theorem. We survey other applications of quadratic forms in design theory and give an original result on the non-existence of certain decompositions of symmetric designs.
29. A Bollobás-type problem: from root systems to Erdos-Ko-Rado, submitted, April 2024.
Joint work with Patrick Browne and Qëndrim Gashi. We analyse a problem of Erdos-Ko-Rado type which appeared in the study of root systems. We give a bound which we show to be almost-tight via an explicit construction using projective planes. We also show that in sufficiently large systems, all optimal solutions are configurations that we term sunflowers, following the EKR literature.
30. Triangle-free graphs with diameter 2, submitted, June 2024.
Joint work with Alice Devillers, Nina Kamčev, Brendan McKay, Gordon Royle, Geertrui Van der Voorde, Ian Wanless and David Wood. This paper resulted from my participation in the MATRIX workshop on extremal problems in geometries, designs and graphs. We explore a relaxation of Moore graphs from combinatorial, computational, algebraic and random perspectives.