Algebraic Combinatorics
http://hdl.handle.net/2160/2971
2014-12-20T05:13:56ZHulls 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, T. P.
The hulls of codes from the row span over F_p, 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 F_p, where A is an adjacency matrix for the graph. If p = 2, for most classes of connected regular graphs with some further form of symmetry, it was shown in P. Dankelmann, J. D. Key, and B. G. Rodrigues, 'Codes from incidence matrices of graphs', Des. Codes Cryptogr. (To appear, 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,lambda,mu), then, unless k is even and mu 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 mu is odd, we show that the binary hull is zero. Further, if a graph is the line graph of a k-regular graph, k>2, that has an l-cycle for some l>2, 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.
2014-01-03T00:00:00ZOn 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, T. P.; Shrikhande, M. S.
In a recent paper, Pawale [22] 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.
2012-04-01T00:00:00ZAn 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 P.; 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.
2009-01-01T00:00:00ZAmalgams of designs and nets
http://hdl.handle.net/2160/5986
Amalgams of designs and nets
McDonough, Thomas P.; 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.
2009-10-17T00:00:00Z