Decomposition of complete designs

Through six chapters, the concept of decomposing the complete design is demonstrated. Group divisible designs, symmetric designs, strongly regular graphs, and association schemes are examples of the combinatorial objects that complete designs are decomposed into. Reconstructing McFarland designs leads to the existence of sets of designs with disjoint incidence matrices whose sum is the complete design. The existence of infinite classes of symmetric association schemes follows from the decomposition. Applying a similar technique on the Spence designs provides sets of designs all sharing the same complete tripartite graphs. By appropriately splitting the designs a decomposition of the complete design is obtained leading to an infinite class of non-commutative association schemes. A final attempt is made to combine the constructed decomposition with specific classes of balanced generalized weighing matrices

Divisible Design Graphs

AMS Subject Classification: 05B05, 05E30, 05C50.Strongly regular graph;Group divisible design;Deza graph;(v;k;)-Graph

Cohomology-Developed Matrices -- constructing families of weighing matrices and automorphism actions

The aim of this work is to construct families of weighing matrices via their automorphism group action. This action is determined from the $0,1,2$-cohomology groups of the underlying abstract group. As a consequence, some old and new families of weighing matrices are constructed. These include the Paley Conference, the Projective-Space, the Grassmannian, and the Flag-Variety weighing matrices. We develop a general theory relying on low dimensional group-cohomology for constructing automorphism group actions, and in turn obtain structured matrices that we call \emph{Cohomology-Developed matrices}. This "Cohomology-Development" generalizes the Cocyclic and Group Developments. The Algebraic structure of modules of Cohomology-Developed matrices is discussed, and an orthogonality result is deduced. We also use this algebraic structure to define the notion of \emph{Quasiproducts}, which is a generalization of the Kronecker-product

Group divisible designs, GBRDSDS and generalized weighing matrices

We give new constructions for regular group divisible designs, pairwise balanced designs, generalized Bhaskar Rao supplementary difference sets and generalized weighing matrices. In particular if p is a prime power and q divides p - 1 we show the following exist; (i) GDD (2(p2+p+1), 2(p2+p+1), rp2,2p2, λ1 = p2λ, λ2 = (p2-p)r, m=p2+p+1,n=2), r_+1,2; (ii) GDD(q(p+1), q(p+1), p(q-1), p(q-1),λ1=(q-1)(q-2), λ2=(p-1)(q-1)2/q,m=q,n=p+1); (iii) PBD(21,10;K),K={3,6,7} and PDB(78,38;K), K={6,9,45}; (iv) GW(vk,k2;EA(k)) whenever a (v,k,λ)-difference set exists and k is a prime power; (v) PBIBD(vk2,vk2,k2,k2;λ1=0,λ2=λ,λ3=k) whenever a (v,k,λ)-difference set exists and k is a prime power; (vi) we give a GW(21;9;Z3)