A feasibility approach for constructing combinatorial designs of circulant type

In this work, we propose an optimization approach for constructing various classes of circulant combinatorial designs that can be defined in terms of autocorrelations. The problem is formulated as a so-called feasibility problem having three sets, to which the Douglas-Rachford projection algorithm is applied. The approach is illustrated on three different classes of circulant combinatorial designs: circulant weighing matrices, D-optimal matrices, and Hadamard matrices with two circulant cores. Furthermore, we explicitly construct two new circulant weighing matrices, a $CW(126,64)$ and a $CW(198,100)$, whose existence was previously marked as unresolved in the most recent version of Strassler's table

Divisible Design Graphs

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

The problem of mutually unbiased bases in dimension 6

We outline a discretization approach to determine the maximal number of mutually unbiased bases in dimension 6. We describe the basic ideas and introduce the most important definitions to tackle this famous open problem which has been open for the last 10 years. Some preliminary results are also listed

Error-Correcting Codes Associated With Generalized Hadamard Matrices Over Groups

Classical Hadamard matrices are orthogonal matrices whose elements are ±1. It is well-known that error correcting codes having large minimum distance between codewords can be associated with these Hadamard matrices. Indeed, the success of early Mars deep-space probes was strongly dependent upon this communication technology. The concept of Hadamard matrices with elements drawn from an Abelian group is a natural generalization of the concept. For the case in which the dimension of the matrix is q and the group consists of the p-th roots of unity, these generalized Hadamard matrices are called “Butson Hadamard Matrices BH(p, q)”, first discovered by A. T. Butson [6]. In this dissertation it is shown that an error correcting code whose codewords consist of real numbers in finite Galois field Gf( p) can be associated in a simple way with each Butson Hadamard matrix BH(p, q), where p \u3e 0 is a prime number. Distance properties of such codes are studied, as well as conditions for the existence of linear codes, for which standard decoding techniques are available. In the search for cyclic linear generalized Hadamard codes, the concept of an M-invariant infinite sequence whose elements are integers in a finite field is introduced. Such sequences are periodic of least period, T, and have the interesting property, that arbitrary identical rearrangements of the elements in each period yields a periodic sequence with the same least period. A theorem characterizing such M-invariant sequences leads to discovery of a simple and efficient polynomial method for constructing generalized Hadamard matrices whose core is a linear cyclic matrix and whose row vectors constitute a linear cyclic error correcting code. In addition, the problem is considered of determining parameter sequences {tn} for which the corresponding potential generalized Hadamard matrices BH(p, ptn) do not exist. By analyzing quadratic Diophantine equations, new methods for constructing such parameter sequences are obtained. These results show the rich number theoretic complexity of the existence question for generalized Hadamard matrices