Constructions and restrictions for balanced splittable Hadamard matrices

TL;DR

This paper studies balanced splittable Hadamard matrices, providing new constraints, classifying their parameters, and constructing infinite families, with implications for related combinatorial and geometric structures.

Contribution

It introduces new combinatorial restrictions, classifies parameter classes based on graph primitivity, and constructs infinite families of balanced splittable Hadamard matrices.

Findings

Parameters are restricted to specific classes via combinatorial analysis.

New infinite families of balanced splittable Hadamard matrices are constructed.

Connections to partial difference sets and elementary abelian 2-groups are established.

Abstract

A Hadamard matrix is balanced splittable if some subset of its rows has the property that the dot product of every two distinct columns takes at most two values. This definition was introduced by Kharaghani and Suda in 2019, although equivalent formulations have been previously studied using different terminology. We collate previous results phrased in terms of balanced splittable Hadamard matrices, real flat equiangular tight frames, spherical two-distance sets, and two-distance tight frames. We use combinatorial analysis to restrict the parameters of a balanced splittable Hadamard matrix to lie in one of several classes, and obtain strong new constraints on their mutual relationships. An important consideration in determining these classes is whether the strongly regular graph associated with the balanced splittable Hadamard matrix is primitive or imprimitive. We construct new infinite families of balanced splittable Hadamard matrices in both the primitive and imprimitive cases. A rich source of examples is provided by packings of partial difference sets in elementary abelian 2-groups, from which we construct Hadamard matrices admitting a row decomposition so that the balanced splittable property holds simultaneously with respect to every union of the submatrices of the decomposition.