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.
@article{10_37236_11586,
author = {Jonathan Jedwab and Shuxing Li and Samuel Simon},
title = {Constructions and restrictions for balanced splittable {Hadamard} matrices},
journal = {The electronic journal of combinatorics},
year = {2023},
volume = {30},
number = {1},
doi = {10.37236/11586},
zbl = {1510.15048},
url = {http://geodesic.mathdoc.fr/articles/10.37236/11586/}
}
TY - JOUR
AU - Jonathan Jedwab
AU - Shuxing Li
AU - Samuel Simon
TI - Constructions and restrictions for balanced splittable Hadamard matrices
JO - The electronic journal of combinatorics
PY - 2023
VL - 30
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/11586/
DO - 10.37236/11586
ID - 10_37236_11586
ER -
%0 Journal Article
%A Jonathan Jedwab
%A Shuxing Li
%A Samuel Simon
%T Constructions and restrictions for balanced splittable Hadamard matrices
%J The electronic journal of combinatorics
%D 2023
%V 30
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/11586/
%R 10.37236/11586
%F 10_37236_11586
Jonathan Jedwab; Shuxing Li; Samuel Simon. Constructions and restrictions for balanced splittable Hadamard matrices. The electronic journal of combinatorics, Tome 30 (2023) no. 1. doi: 10.37236/11586
