Algorithms for difference families in finite abelian groups

Our main objective is to show that the computational methods that we previously developed to search for difference families in cyclic groups can be fully extended to the more general case of arbitrary finite abelian groups. In particular the power density PSD-test and the method of compression can be used to help the search.Comment: 18 pages, minor change

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

Application of the Discrete Fourier Transform to the Search for Generalised Legendre Pairs and Hadamard Matrices

We introduce Legendre sequences and generalised Legendre pairs (GL--pairs). We show how to construct an Hadamard matrix of order 2` + 2 from a GL--pair of length `. We review the known constructions for GL--pairs and use the discrete Fourier transform (DFT) and power spectral density (PSD) to enable an exhaustive search for GL--pairs for lengths ` 45 and partial results for other `. 1 Definitions and Notation Let U be a sequence of ` real numbers u 0 ; u 1 ; :::; u `\Gamma1 . The periodic autocorrelation function PU (j) of such a sequence is defined by: PU (j) = `\Gamma1 X i=0 u i u i+j mod ` ; j = 0; 1; :::; ` \Gamma 1: Two sequences U and V of identical length ` are said to be compatible if the sum of their periodic autocorrelations is a constant, say a, except for the 0-th term. That is, PU (j) + P V (j) = a; j 6= 0: (1) (Such pairs are said to have constant periodic autocorrelation even though it is the sum of the autocorrelations that is a constant.) If U and V are both \Si..

Cocyclic Hadamard Matrices: An Efficient Search Based Algorithm

This dissertation serves as the culmination of three papers. “Counting the decimation classes of binary vectors with relatively prime fixed-density presents the first non-exhaustive decimation class counting algorithm. “A Novel Approach to Relatively Prime Fixed Density Bracelet Generation in Constant Amortized Time presents a novel lexicon for binary vectors based upon the Discrete Fourier Transform, and develops a bracelet generation method based upon the same. “A Novel Legendre Pair Generation Algorithm expands upon the bracelet generation algorithm and includes additional constraints imposed by Legendre Pairs. It further presents an efficient sorting and comparison algorithm based upon symmetric functions, as well as multiple unique Legendre Pairs