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William J. Martin
Stratton Hall 302A
William J. Martin
Bioinformatics and Computational Biology
BA State University of New York Potsdam 1986
MA State University of New York Potsdam 1986
PhD University of Waterloo 1992
Bill Martin's goal is to find mathematical research projects that lie between beautiful and powerful mathematical theory, on the one hand, and pressing technological applications, on the other. This effort requires one to keep abreast of both mathematical developments and applications in computer science and engineering. Professor Martin's mathematical research is in the area of algebraic combinatorics, where tools from linear and abstract algebra are applied to problems in discrete math. An association scheme is a collection of graphs, which give rise to a highly structured matrix algebra whose eigenspaces reveal information about these graphs and their substructures. The vertices of the graphs might, for example, be the set of all binary n-tuples in which case we have a tool for the study of error-correcting codes. In this and numerous other cases, by embedding unstructured configurations into well-structured ambient spaces, we obtain algebraic leverage over what are otherwise messy applied problems. Martin and co-authors have applied the theory of association schemes to the study of experimental designs, finite geometries, highly regular graphs, error-correcting codes, (t,m,s)-nets, and structures appearing in quantum information theory. Martin's current research activities are split across four areas. With Professor Berk Sunar, Martin is currently investigating homomorphic encryption, exploring techniques for efficient implementation of existing schemes as well as developing entirely new schemes. With his collaborators, he is carrying out research on mutually unbiased bases. He also works with co-authors in Korea on completely regular error-correcting codes and their connection to distance-regular graphs. Finally, he also uses algebraic and combinatorial techniques to develop association scheme theory itself. In addition to these main activities, Professor Martin is interested in K-12 education, contributing to math clubs, competitions, summer camps, and high school curricular development.
Stratton Hall 302A
Symmetric designs, sets with two intersection numbers, and Krein parameters of incidence graphs - 2001
A new notion of transitivity for groups and sets of permutations - 2006
On the ideal of the shortest vectors in the Leech lattice and other lattices - 2015
Completely regular designs of strength one - 1994
Anticodes for the Grassman [sic] and bilinear forms graphs - 1995
Completely regular designs - 1998
Mixed block designs - 1998
Association schemes for ordered orthogonal arrays and (T,M,S)-nets. - 1999
Designs in product association schemes - 1999
``Arithmetic completely regular codes'' (with Jack Koolen, Woosun Lee, and Hajime Tanaka), Discrete Mathematics and Theoretical Computer Science 17 no. 3 (2016), 59--76
`Bounded, yet sufficient? How to determine whether limited side channel information enables key recovery'' (with T.~Eisenbarth and X.~Ye), (preprint) in: Smart Card Research and Advanced Applications Volume 8968 of the series Lecture Notes in Computer Science pp 215-232
``Linking systems in nonelementary abelian groups'' (with Jim Davis and John Polhill). J. Combin. Theory Ser. A 123, no. 1 (2014), pp. 92-103
``Uniformity in association schemes and coherent configurations: cometric Q-antipodal schemes and linked systems'' (with Edwin van Dam and Mikhail Muzychuk). Journal of Combinatorial Theory, Series A 120, Issue 7, September 2013, Pages 1401--1439
``Enhanced Flexibility for Homomorphic Encryption Schemes via CRT '' (with Yin Hu and Berk Sunar). Industrial Track of ACNS '12, 10th International Conference on Applied Cryptography and Network Security , June 26-29, 2012, Singapore.
``Characterizing completely regular codes from an algebraic viewpoint'' (with Jack Koolen and Woosun Lee), arXiv:0911.1828 in: Combinatorics and Graphs Contemporary Mathematics (Vol. 531) Richard A. Brualdi, Samad Hedayat, Hadi Kharaghani, Gholamreza B. Khosrovshahi, and Shahriar Shahriari, editors.
``On the equivalence between real mutually unbiased bases and a certain class of association schemes'' (with Nick LeCompte and Will Owens). European Journal of Combinatorics 31, no. 6 (2010), pp. 1499-1512. DOI link: http://dx.doi.org/10.1016/j.ejc.2009.11.014
``Resilient functions: Just how resilient are the they?'' (with Berk Sunar), in: Error-Correcting Codes, Finite Geometries and Cryptography Contemporary Mathematics, Vol. 523 Aiden A. Bruen, and David L. Wehlau, editors. Published by the American Mathematical Society, 2010.
``Commutative association schemes'' (with Hajime Tanaka). European Journal of Combinatorics 30, no. 6 (2009), pp. 1497-1525. DOI link: http://dx.doi.org/10.1016/j.ejc.2008.11.001
``There are finitely many Q-polynomial association schemes with given first multiplicity at least three.'' (with Jason Williford). European Journal of Combinatorics 30, no. 3 (2009), pp. 698-704. DOI link: http://dx.doi.org/10.1016/j.ejc.2008.07.009
``Representations of directed strongly regular graphs.'' (with C. D. Godsil and Sylvia Hobart.) European Journal of Combinatorics 28 no. 7 (2007), 1980--1993. (Proceedings of "Geometric and Algebraic Combinatorics 3".)
``Imprimitive cometric association schemes: constructions and analysis.'' (with Mikhail Muzychuk and Jason Williford). Journal of Algebraic Combinatorics 25 no. 4 (2007), 399--415.
``A provably secure true random number generator with built-in tolerance to active attacks.'' (with B. Sunar and D.R. Stinson) IEEE Transactions on Computers 56 no. 1 (2007), 109--119.
``A dual Plotkin bound for (T,M,S)-nets'' (with Terry Visentin) IEEE Trans. Inform. Theory, 53, no. 1 (2007), 411-415..
``Some new constructions of imprimitive cometric association schemes.'' (with Mikhail Muzychuk and Jason Williford). Proceedings, "Algebraic Combinatorics", an international conference in honor of Eiichi Bannai's 60th birthday, June 26-30, 2006, Sendai International Center, Sendai, Japan (not refereed).
``Linear programming bounds for $(T,M,S)$-nets'' (extended abstract), Proceedings of the Seventh Asian Symposium on Computer Mathematics, Seoul, Republic of Korea, December 2005.
`A new notion of transitivity for groups and sets of permutations.'' (with B. E. Sagan) Journal of the London Mathematical Society, 73 (2006), 1-13 .
`Completely regular codes: a viewpoint and some problems.'' pages 43--56 in: Proceedings of 2004 Com2MaC Workshop on Distance-Regular Graphs and Finite Geometry, July 24 - 26, 2004, Pusan, Korea (invited, not refereed).
``A physics-free introduction to quantum error-correcting codes.'' Utilitas Mathematica, 65 (2004), 133-158.
`Width and dual width of subsets in polynomial association schemes.'' (with A. E. Brouwer, C. D. Godsil and J. Koolen) Journal of Combinatorial Theory, Series A 102 (2003), 255-271.
``Symmetric designs, sets with two intersection numbers, and Krein parameters of incidence graphs'' Journal of Combin. Math. and Combin. Computing 38 (2001), 185-196.
``Design systems: combinatorial characterizations of Delsarte T-designs via partially ordered sets.'' pp. 223-239, in: Codes and Association Schemes DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 56, (A. Barg and S. Litsyn, eds.) American Mathematical Society, 2001.
``Minimum distance bounds for s-regular codes.'' Designs, Codes, and Crypt. 21, Issue 1/3 (2000), 181-187. (Proceedings of the conference ``GEOMETRIC AND ALGEBRAIC COMBINATORICS'' in honor of the 80th birthday of J.J. Seidel, August 15-20, 1999.)
``Linear programming bounds for ordered orthogonal arrays and (T,M,S)-nets.'' pp. 368--376, in: Monte Carlo and Quasi-Monte Carlo Methods 1998: Proceedings of a Conference held at the Claremont Graduate University, Claremont, California, USA, June 22-26, 1998. H. Niederreiter and J. Spanier, eds.
``Association schemes for ordered orthogonal arrays and (T,M,S)-nets.'' (with D.R. Stinson), Canad. J. Math. 51, no. 2 (1999), 326-346.
``A generalized Rao bound for ordered orthogonal arrays and (t,m,s)-nets.'' (with D.R. Stinson), Canadian Math. Bulletin 42, no. 3 (1999), 359-370.
``Anticodes for the Grassman [sic] and bilinear forms graphs,'' (with X.J. Zhu) Designs, Codes, and Crypt. 6, no. 1 (1995), 73-79.
``Quotients of association schemes,'' (with C.D. Godsil) J. Combin. Th., Ser. A 69, no. 2 (1995), 185-199.