Modular Hadamard matrices and related designs,III

This paper continues the investigations presented in two previous papers on the same subject by the author and A. T. Butson. Modular Hadamard matrices havingn odd andh ≡ ? 1 (modn) are studied for a few values of the parametersn andh. Also, some results are obtained for the two related combinatorial designs. These results include: a discussion on the known techniques for constructing pseudo (v, k, λ)-designs; the fact that the existence of one of the two related designs always implies the existence of the other; and some information about the column sums of the incidence matrix of each of the two ‘maximal’ cases of (m, v, k ₁,λ ₁,k ₂,λ ₂,f, λ ₃)-designs.     

Hyperovals and Hadamard designs

In this paper, we investigatec-sets in 2-designs, with particular regard to sets of type (0,n) in projective planes. In particular, we associate a Hadamard design to a hyperoval of a projective plane of even orderq and we investigate some properties of its lines. This gives information on the order of the projective plane.     

Nonisomorphic Hadamard designs

A block b of a Hadamard design is called a good block if the symmetric difference b + b₁ is also a block for all nonparallel blocks b₁. The isomorphism classes of such designs having a good block are shown to be related to a double coset decomposition of a symmetric group. As an example, over one million mutually nonisomorphic 3-(32, 16, 7) designs of a certain type are constructed.Equivalence of Hadamard matrices is described in terms of designs and it is shown that nonisomorphic designs may arise from the same matrix.     

Enumeration of generalized Hadamard matrices of order 16 and related designs

We investigate signings of symmetric GDD( , 16, )s over for . Beginning with , at each stage of this process a signing of a GDD( , 16, ) produces a GDD( , 16, ). The initial GDDs ( ) correspond to Hadamard matrices of order 16. For , the GDDs are semibiplanes of order 16, and for the GDDs are semiplanes of order 16 which can be extended to projective planes of order 16. In this article, we completely enumerate such signings which include all generalized Hadamard matrices of order 16. We discuss the generation techniques and properties of the designs obtained during the search. © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 119–135, 2009     

Hadamard Matrices Modulo p and Small Modular Hadamard Matrices

We use modular symmetric designs to study the existence of Hadamard matrices modulo certain primes. We solve the 7‐modular and 11‐modular versions of the Hadamard conjecture for all but a finite number of cases. In doing so, we state a conjectural sufficient condition for the existence of a p‐modular Hadamard matrix for all but finitely many cases. When 2 is a primitive root of a prime p, we conditionally solve this conjecture and therefore the p‐modular version of the Hadamard conjecture for all but finitely many cases when , and prove a weaker result for . Finally, we look at constraints on the existence of m‐modular Hadamard matrices when the size of the matrix is small compared to m.     

Modular sequences and modular Hadamard matrices

For every n divisible by 4, we construct a square matrix H of size n, with coefficients ± 1, such that H · H^t ≡ nI mod 32. This solves the 32‐modular version of the classical Hadamard conjecture. We also determine the set of lengths of 16‐modular Golay sequences. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 187–214, 2001     

On affine designs and Hadamard designs with line spreads

Rahilly [On the line structure of designs, Discrete Math. 92 (1991) 291-303] described a construction that relates any Hadamard design H on 4^m-1 points with a line spread to an affine design having the same parameters as the classical design of points and hyperplanes in AG(m,4). Here it is proved that the affine design is the classical design of points and hyperplanes in AG(m,4) if, and only if, H is the classical design of points and hyperplanes in PG(2m-1,2) and the line spread is of a special type. Computational results about line spreads in PG(5,2) are given. One of the affine designs obtained has the same 2-rank as the design of points and planes in AG(3,4), and provides a counter-example to a conjecture of Hamada [On the p-rank of the incidence matrix of a balanced or partially balanced incomplete block design and its applications to error-correcting codes, Hiroshima Math. J. 3 (1973) 153-226].     

Skew Hadamard designs and their codes

Skew Hadamard designs (4n – 1, 2n – 1, n – 1) are associated to order 4n skew Hadamard matrices in the natural way. We study the codes spanned by their incidence matrices A and by I + A and show that they are self-dual after extension (resp. extension and augmentation) over fields of characteristic dividing n. Quadratic Residues codes are obtained in the case of the Paley matrix. Results on the p-rank of skew Hadamard designs are rederived in that way. Codes from skew Hadamard designs are classified. An optimal self-dual code over GF(5) is rediscovered in length 20. Six new inequivalent [56, 28, 16] self-dual codes over GF(7) are obtained from skew Hadamard matrices of order 56, improving the only known quadratic double circulant code of length 56 over GF(7).     

Structure and automorphism groups of Hadamard designs

Let n be the order of a Hadamard design, and G any finite group. Then there exists many non-isomorphic Hadamard designs of order 2^{12|G| + 13} n with automorphism group isomorphic to G.This research was supported in part by the National Science Foundation.     

On Hadamard designs associated with a hyperoval

Hadamard designs which can be associated with a hyperoval of a projective plane of even order are investigated. In particular, when is a translation hyperoval, these designs are shown to contain restrictions that are isomorphic to the 2-design of points and hyperplanes of a projective geometry overGF(2).     

(19, 9, 4) Hadamard designs and their residual designs

In this paper we study (19, 9, 4) Hadamard designs and their residual designs. We prove that there are precisely six non-isomorphic solutions of (19, 9, 4) designs and that these six designs give rise to in all twenty-one mutually non-isomorphic residual designs.     

Bush‐type Hadamard matrices and symmetric designs

Abstact: A symmetric 2‐(100, 45, 20) design is constructed that admits a tactical decomposition into 10 point and block classes of size 10 such that every point is in either 0 or 5 blocks from a given block class, and every block contains either 0 or 5 points from a given point class. This design yields a Bush‐type Hadamard matrix of order 100 that leads to two new infinite classes of symmetric designs with parameters and where m is an arbitrary positive integer. Similarly, a Bush‐type Hadamard matrix of order 36 is constructed and used for the construction of an infinite family of designs with parameters and a second infinite family of designs with parameters where m is any positive integer. © 2000 John Wiley & Sons, Inc. J Combin Designs 9: 72–78, 2001     

Modified generalized Hadamard matrices and constructions for transversal designs

It is well known that there exists a transversal design TD_λ[k; u] admitting a class regular automorphism group U if and only if there exists a generalized Hadamard matrix GH(u, λ) over U. Note that in this case the resulting transversal design is symmetric by Jungnickel’s result. In this article we define a modified generalized Hadamard matrix and show that transversal designs which are not necessarily symmetric can be constructed from these under a modified condition similar to class regularity even if it admits no class regular automorphism group.     

Some Hadamard designs with parameters (71,35,17)

Up to isomorphisms there are precisely eight symmetric designs with parameters (71, 35, 17) admitting a faithful action of a Frobenius group of order 21 in such a way that an element of order 3 fixes precisely 11 points. Five of these designs have 84 and three have 420 as the order of the full automorphism group G. If |G| = 420, then the structure of G is unique and we have G = (Frob₂₁ × Z5):Z4. In this case Z(G) = 〈1〉, G′ has order 35, and G induces an automorphism group of order 6 of Z7. If |G| = 84, then Z(G) is of order 2, and in precisely one case a Sylow 2‐subgroup is elementary abelian. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 144–149, 2002; DOI 10.1002/jcd.996     

On symmetric nets and generalized Hadamard matrices from affine designs

Symmetric nets are affine resolvable designs whose duals are also affine. It is shown that. up to isomorphism, there are exactly four symmetric (3, 3)-nets (v=b=27,k=9), and exactly two inequivalent 9×9 generalized Hadamard matrices over the group of order 3. The symmetric (3, 3)-nets are found as subnets of affine resolvable 2-(27, 9, 4) designs. Ten of the 68 non-isomorphic affine resolvable 2-(27, 9, 4) designs are not extensions of symmetric (3, 3)-subnets, providing the first examples of affine 2-(q³, q², q²–1/q–1) designs without symmetric (q, q)-subnets.