A feasibility approach for constructing combinatorial designs of circulant type

TL;DR

This paper introduces an optimization-based feasibility approach using the Douglas-Rachford algorithm to construct circulant combinatorial designs, successfully creating new matrices and solving previously unresolved cases.

Contribution

It presents a novel optimization method for constructing circulant combinatorial designs and explicitly constructs two new unresolved circulant weighing matrices.

Findings

Successfully constructed new circulant weighing matrices CW(126,64) and CW(198,100)

Applied the approach to three classes of designs: weighing, D-optimal, and Hadamard matrices

Demonstrated the method's effectiveness in solving unresolved design existence problems

Abstract

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.