Algebraic Combinatorics
http://hdl.handle.net/2160/2971
2016-05-01T04:20:16ZAmalgams of designs and nets
http://hdl.handle.net/2160/33783
Amalgams of designs and nets
McDonough, Thomas; Ward, H. N.; Mavron, V. C.
We present a procedure for amalgamating a net and a collection of designs into a single design. At first this amalgam is just point-regular, but it acquires additional regularities upon imposing restrictions on the ingredients. At its most regular, the amalgam is quasi-symmetric, and designs with the same parameters as those recently constructed by Bracken, McGuire and Ward appear. Along the way we discuss a class of designs generalising Hadamard designs, and we consider the problem of packing projective planes with disjoint line sets into the same point set.
T.P.McDonough, V.C.Mavron and H.N.Ward, Amalgams of designs and nets. Bulletin of the London Mathematical Society (2009) 41 (5): 841-852.
2009-10-17T00:00:00ZReed-Muller codes and permutation decoding
http://hdl.handle.net/2160/32339
Reed-Muller codes and permutation decoding
Key, Jennifer D.; McDonough, Thomas; Mavron, Vassili C.
We show that the first- and second-order Reed–Muller codes, and , can be used for permutation decoding by finding, within the translation group, (m−1)- and (m+1)-PD-sets for for m≥5,6, respectively, and (m−3)-PD-sets for for m≥8. We extend the results of Seneviratne [P. Seneviratne, Partial permutation decoding for the first-order Reed-Muller codes, Discrete Math., 309 (2009), 1967–1970].
Mavron, V. C., Key, J. D., McDonough, T. P. (2010) Reed-Muller codes and permutation decoding. Discrete Mathematics, 310 (22), 3114-3119 Sponsorship: London Mathematical Society (Partial support)
2010-11-28T00:00:00ZOn subsequences and certain elements which determine various cells in S_n
http://hdl.handle.net/2160/32336
On subsequences and certain elements which determine various cells in S_n
Pallikaros, C. A.; McDonough, Thomas
We study the relation between certain increasing and decreasing subsequences occurring in the row form of certain elements in the symmetric group, following Schensted [C. Schensted, Longest increasing and decreasing subsequences, Canad. J. Math. 13 (1961) 179–191] and Greene [C. Greene, An extension of Schensted's theorem, Adv. Math. 14 (1974) 254–265], and the Kazhdan–Lusztig cells [D.A. Kazhdan, G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979) 165–184] of the symmetric group to which they belong. We show that, in the two-sided cell corresponding to a partition λ, there is an explicitly defined element dλ, each of whose prefixes can be used to obtain a left cell by multiplying the cell containing the longest element of the parabolic subgroup associated with λ on the right. Furthermore, we show that the elements of these left cells are those which possess increasing and decreasing subsequences of certain types. The results in this paper lead to efficient algorithms for the explicit descriptions of many left cells inside a given two-sided cell, and the authors have implemented these algorithms in GAP.
T.P.McDonough, C.A.Pallikaros, On subsequences and certain elements which determine various cells in S_n, Journal of Algebra, 2008, volume 319, issue 3, pp. 1249-1263.
2008-02-01T00:00:00ZPartial permutation decoding for codes from affine geometry designs
http://hdl.handle.net/2160/31886
Partial permutation decoding for codes from affine geometry designs
McDonough, Thomas; Key, Jennifer D.; Mavron, Vassili
Explicit PD-sets are found for partial permutation decoding of the generalized Reed-Muller codes from the affine geometry designs of points and lines in dimension 3 over the prime field of order p, using the information sets described by the authors in an earlier paper.
Journal of Geometry, 88, No. 1-2, 2008, 101-109. DOI: 10.1007/s00022-007-1922-y
2008-03-01T00:00:00Z