Algebraic Combinatorics http://hdl.handle.net/2160/2971 2020-04-04T18:48:01Z Classification of book spreads in PG(5, 2) http://hdl.handle.net/2160/30066 Classification of book spreads in PG(5, 2) McDonough, Thomas; Shaw, R.; Topalova, S. We classify all line spreads S21 in PG(5, 2) of a special kind, namely those which are book spreads. We show that up to isomorphism there are precisely nine different kinds of book spreads and describe the automorphism groups which stabilize them. Most of the main results are obtained in two independent ways, namely theoretically and by computer. McDonough, T., Shaw, R., Topalova, S. (2013). Classification of book spreads in PG(5, 2). Note di Matematica, 33 (2), 43-64 2013-01-01T00:00:00Z Hulls of codes from incidence matrices of connected regular graphs http://hdl.handle.net/2160/7822 Hulls of codes from incidence matrices of connected regular graphs Ghinelli, D.; Key, J. D.; McDonough, Thomas The hulls of codes from the row span over FpFp , for any prime p, of incidence matrices of connected k-regular graphs are examined, and the dimension of the hull is given in terms of the dimension of the row span of A + kI over FpFp , where A is an adjacency matrix for the graph. Ifp = 2, for most classes of connected regular graphs with some further form of symmetry, it was shown by Dankelmann et al. (Des. Codes Cryptogr. 2012) that the hull is either {0} or has minimum weight at least 2k−2. Here we show that if the graph is strongly regular with parameter set (n, k, λ, μ), then, unless k is even and μ is odd, the binary hull is non-trivial, of minimum weight generally greater than 2k − 2, and we construct words of low weight in the hull; if k is even and μ is odd, we show that the binary hull is zero. Further, if a graph is the line graph of a k-regular graph, k ≥ 3, that has an ℓ-cycle for some ℓ ≥ 3, the binary hull is shown to be non-trivial with minimum weight at most 2ℓ(k−2). Properties of the p-ary hulls are also established. D.Ghinelli, J.D.Key, T.P.McDonough. Hulls of codes from incidence matrices of connected regular graphs. Designs Codes and Cryptography, 2014, vol 70, pg 35-54. 2014-01-01T00:00:00Z On quasi-symmetric designs with intersection difference three http://hdl.handle.net/2160/7800 On quasi-symmetric designs with intersection difference three Mavron, V. C.; McDonough, Thomas; Shrikhande, M. S. In a recent paper, Pawale  investigated quasi-symmetric 2-(v, k, lambda) designs with intersection numbers x > 0 and y = x + 2 with lambda > 1 and showed that under these conditions either lambda = x + 1 or lambda = x + 2, or D is a design with parameters given in the form of an explicit table, or the complement of one of these designs. In this paper, quasi-symmetric designs with y-x = 3 are investigated. It is shown that such a design or its complement has parameter set which is one of finitely many which are listed explicitly or lambda V.C. Mavron, T.P. McDonough and M.S. Shrikhande (2012). On quasi-symmetric designs with intersection difference three. Designs, Codes and Cryptography, 63 (1), 73-86. 2012-04-01T00:00:00Z An upper bound for the minimum weight of the dual codes of desarguesian planes http://hdl.handle.net/2160/5987 An upper bound for the minimum weight of the dual codes of desarguesian planes Mavron, V. C.; McDonough, Thomas; Key, Jennifer D. We show that a construction described in [K.L. Clark, J.D. Key, M.J. de Resmini, Dual codes of translation planes, European J. Combinatorics 23 (2002) 529–538] of small-weight words in the dual codes of finite translation planes can be extended so that it applies to projective and affine desarguesian planes of any order p^m where p is a prime, and m≥1. This gives words of weight 2p^m+1-(p^m-1)/(p-1) in the dual of the p-ary code of the desarguesian plane of order p^m, and provides an improved upper bound for the minimum weight of the dual code. The same will apply to a class of translation planes that this construction leads to; these belong to the class of André planes. We also found by computer search a word of weight 36 in the dual binary code of the desarguesian plane of order 32, thus extending a result of Korchmáros and Mazzocca [Gábor Korchmáros, Francesco Mazzocca, On (q+t)-arcs of type (0,2,t) in a desarguesian plane of order q, Math. Proc. Cambridge Phil. Soc. 108 (1990) 445–459]. J.D.Key, T.P.McDonough and V.C.Mavron, An upper bound for the minimum weight of the dual codes of desarguesian planes. European Journal of Combinatorics, Volume 30 Issue 1, January, 2009, pp. 220-229. 2009-01-01T00:00:00Z