The codes and the lattices of Hadamard matrices

It has been observed by Assmus and Key as a result of the complete classification of Hadamard matrices of order 24, that the extremality of the binary code of a Hadamard matrix H of order 24 is equivalent to the extremality of the ternary code of H^T. In this note, we present two proofs of this fact, neither of which depends on the classification. One is a consequence of a more general result on the minimum weight of the dual of the code of a Hadamard matrix. The other relates the lattices obtained from the binary code and from the ternary code. Both proofs are presented in greater generality to include higher orders. In particular, the latter method is also used to show the equivalence of (i) the extremality of the ternary code, (ii) the extremality of the Z_4-code, and (iii) the extremality of a lattice obtained from a Hadamard matrix of order 48.Comment: 16 pages. minor revisio

Finding D-optimal designs by randomised decomposition and switching

A square {+1,-1}-matrix of order n with maximal determinant is called a saturated D-optimal design. We consider some cases of saturated Doptimal designs where n > 2, n ≢ 0 mod 4, so the Hadamard bound is not attainable, but bounds due to Barba or Ehlic

Hadamard 2-(63,31,15) designs invariant under the dihedral group of order 10

AbstractAll Hadamard 2-(63,31,15) designs invariant under the dihedral group of order 10 are constructed and classified up to isomorphism together with related Hadamard matrices of order 64. Affine 2-(64,16,5) designs can be obtained from Hadamard 2-(63,31,15) designs having line spreads by Rahilly’s construction [A. Rahilly, On the line structure of designs, Discrete Math. 92 (1991) 291–303]. The parameter set 2-(64,16,5) is one of two known sets when there exists several nonisomorphic designs with the same parameters and p-rank as the design obtained from the points and subspaces of a given dimension in affine geometry AG(n,pm) (p a prime). It is established that an affine 2-(64,16,5) design of 2-rank 16 that is associated with a Hadamard 2-(63,31,15) design invariant under the dihedral group of order 10 is either isomorphic to the classical design of the points and hyperplanes in AG(3,4), or is one of the two exceptional designs found by Harada, Lam and Tonchev [M. Harada, C. Lam, V.D. Tonchev, Symmetric (4, 4)-nets and generalized Hadamard matrices over groups of order 4, Designs Codes Cryptogr. 34 (2005) 71–87]

Algorithms for classification of combinatorial objects

A recurrently occurring problem in combinatorics is the need to completely characterize a finite set of finite objects implicitly defined by a set of constraints. For example, one could ask for a list of all possible ways to schedule a football tournament for twelve teams: every team is to play against every other team during an eleven-round tournament, such that every team plays exactly one game in every round. Such a characterization is called a classification for the objects of interest. Classification is typically conducted up to a notion of structural equivalence (isomorphism) between the objects. For example, one can view two tournament schedules as having the same structure if one can be obtained from the other by renaming the teams and reordering the rounds. This thesis examines algorithms for classification of combinatorial objects up to isomorphism. The thesis consists of five articles – each devoted to a specific family of objects – together with a summary surveying related research and emphasizing the underlying common concepts and techniques, such as backtrack search, isomorphism (viewed through group actions), symmetry, isomorph rejection, and computing isomorphism. From an algorithmic viewpoint the focus of the thesis is practical, with interest on algorithms that perform well in practice and yield new classification results; theoretical properties such as the asymptotic resource usage of the algorithms are not considered. The main result of this thesis is a classification of the Steiner triple systems of order 19. The other results obtained include the nonexistence of a resolvable 2-(15, 5, 4) design, a classification of the one-factorizations of k-regular graphs of order 12 for k ≤ 6 and k = 10, 11, a classification of the near-resolutions of 2-(13, 4, 3) designs together with the associated thirteen-player whist tournaments, and a classification of the Steiner triple systems of order 21 with a nontrivial automorphism group.reviewe

Amicable matrices and orthogonal designs

This thesis is mainly concerned with the orthogonal designs of Baumert-Hall array type, OD(4n;n,n,n,n) where n=2k, k is odd integer. For every odd prime power p^r, we construct an infinite class of amicable T-matrices of order n=p^r+1 in association with negacirculant weighing matrices W(n,n-1). In particular, for p^r≡1 (mod 4) we construct amicable T-matrices of order n≡2 (mod 4) and application of these matrices allows us to generate infinite class of orthogonal designs of type OD(4n;n,n,n,n) and OD(4n;n,n,n-2,n-2) where n=2k; k is odd integer. For a special class of T-matrices of order n where each of T_i is a weighing matrix of weight w_i;1 ≤i≤4 and Williamson-type matrices of order m, we establish a theorem which produces four circulant matrices in terms of four variables. These matrices are additive and can be used to generate a new class of orthogonal design of type OD(4mn;w_1s,w_2s,w_3s,w_4s ); where s=4m. In addition to this, we present some methods to find amicable matrices of odd order in terms of variables which have an interesting application to generate some new orthogonal designs as well as generalized orthogonal designs.University of Lethbridge, NSER