Balanced generalized weighing matrices and their applications

Balanced generalized weighing matrices include well-known classical combinatorial objects such as Hadamard matrices and conference matrices; moreover, particular classes of BGW -matrices are equivalent to certain relative difference sets. BGW -matrices admit an interesting geometrical interpretation, and in this context they generalize notions like projective planes admitting a full elation or homology group. After surveying these basic connections, we will focus attention on proper BGW -matrices; thus we will not give any systematic treatment of generalized Hadamard matrices, which are the subject of a large area of research in their own right. In particular, we will discuss what might be called the classical parameter series. Here the nicest examples are closely related to perfect codes and to some classical relative difference sets associated with affine geometries; moreover, the matrices in question can be characterized as the unique (up to equivalence) BGW -matrices for the given parameters with minimum q-rank.One can also obtain a wealth of monomially inequivalent examples and deter  mine the q-ranks of all these matrices by exploiting a connection with linear shift register sequences

On orthogonal matrices

iv, 64 leaves : ill., map ; 29 cm.Our main aim in this thesis is to study and search for orthogonal matrices which have a certain kind of block structure. The most desirable class of matrices for our purpose are orthogonal designs constructible from 16 circulant matrices. In studying these matrices, we show that the OD (12;1,1,1,9) is the only orthogonal design constructible from 16 circulant matrices of type OD (4n;1,1,1,4n-3), whenever n > 1 is an odd integer. We then use an exhaustive search to show that the only orthogonal design constructible from 16 circulant matrices of order 12 on 4 variables is the OD (12;1,1,1,9). It is known that by using of T-matrices and orthogonal designs constructible from 16 circulant matrices one can produce an infinite family of orthogonal designs. To complement our studies we reproduce and important recent construction of T-matrices by Xia and Xia. We then turn our attention to the applications of orthogonal matices. In some recent works productive regular Hadamard matrices are used to construct many new infinite families of symmetric designs. We show that for each integer n for which 4n is the order of a Hadamard matrix and 8n2 - 1 is a prime, there is a productive regular Hadamard matrix of order 16n2(82-1)2. As a corollary, we get many new infinite classes of symmetric designs whenever either of 4n(8n2-1)-1,4n(82-1) +1 is a prime power. We also review some other constructions of productive regular Hadamard matrices which are related to our work

Coding Theory and Algebraic Combinatorics

This chapter introduces and elaborates on the fruitful interplay of coding theory and algebraic combinatorics, with most of the focus on the interaction of codes with combinatorial designs, finite geometries, simple groups, sphere packings, kissing numbers, lattices, and association schemes. In particular, special interest is devoted to the relationship between codes and combinatorial designs. We describe and recapitulate important results in the development of the state of the art. In addition, we give illustrative examples and constructions, and highlight recent advances. Finally, we provide a collection of significant open problems and challenges concerning future research.Comment: 33 pages; handbook chapter, to appear in: "Selected Topics in Information and Coding Theory", ed. by I. Woungang et al., World Scientific, Singapore, 201