Isolated Partial Hadamard Matrices and Related Topics

Abstract

We study the isolated partial Hadamard matrices, under the assumption that the entries are roots of unity, or more generally, under the assumption that the combinatorics comes from vanishing sums of roots of unity. We first review the various conjectures on the subject, and then we present several new results, regarding notably the master Hadamard matrices, and the McNulty-Weigert construction. We discuss then the notion of isolation in some related contexts, of the magic unitary matrices, and of the quantum permutation groups, with a number of conjectures on the subject.2010 Mathematics Subject Classification. 15B34 (46L65, 81R50).

Invitation to Hadamard matrices

“…We should mention that the above result, as well as the whole basic theory of the tangent cones and defect, can be extended in a quite straightforward way to the case of the partial Hadamard matrices [22]. We will be back to this, later on.…”

“…Besides the classification questions, there are as well a number of theoretical questions in relation with the notion of regularity, that we believe to be very interesting. We have for instance the following conjecture, going back to [13], and then to [22]: Conjecture 3.22 (Regularity Conjecture). The following hold:…”

“…As for the second conjecture, this simply comes from the known examples of regular Hadamard matrices, which all appear from certain Butson matrices, by inserting parameters, in an affine way. This conjecture is from [22], and we will further discuss the notion of affine deformation, with some general results on the subject, in section 4 below.…”

“…We discuss here a number of more specialized aspects, regarding the tensor products, the Dit ¸ȃ deformations of such tensor products, the Butson and the regular matrices, the master Hadamard matrices, the McNulty-Weigert matrices, and finally the partial Hadamard matrix case, following [2], [5], [6], [22], [69], [88], [89].…”

Pseudococyclic Partial Hadamard Matrices over Latin Rectangles

“…We remind the reader in this regard that a partial Hadamard matrix is an r × n (binary) matrix H with r ≤ n such that HH t = nI r . The recent implementation of these types of matrices in cryptography [19], experimental design [20], and quantum information [21] has awakened the interest in describing different ways of constructing them [22][23][24][25]. In addition, Latin rectangles may be implemented in Internet of Things (IoT) studies [26], coding theory [27,28], and modern 5G wireless networks [29].…”