The quaternary complex Hadamard matrices of orders 10, 12, and 14

Abstract

A complete classification of quaternary complex Hadamard matrices of orders 10, 12 and 14 is given, and a new parametrization scheme for obtaining new examples of affine parametric families of complex Hadamard matrices is provided. On the one hand, it is proven that all 10 × 10 and 12 × 12 quaternary complex Hadamard matrices belong to some parametric family, but on the other hand, it is shown by exhibiting an isolated 14 × 14 matrix that there cannot be a general method for introducing parameters into these types of matrices.the BH(q, n) matrices makes it possible to escape equivalence classes and therefore to collect many inequivalent matrices into a single parametric family. Also the switching operation [16], a well-known technique in design theory, has this property.The existence of BH(q, n) matrices is wide open in general. Even the simplest case for q = 2 is undecided, as the famous Hadamard conjecture, stating that BH(2, 4k) matrices exist for every positive integer k, remains elusive despite continuous efforts [3]. Real Hadamard matrices, or BH(2, n) matrices, are completely classified up to n = 28, and there has been some promising advance in enumerating the case n = 32 very recently (see [7] and the references therein). For other values of q we have some constructions [2] and some nonexistence results [20]. Harada, Lam and Tonchev classified all 16 × 16 generalized Hadamard matrices over groups of order 4 and obtained new examples of BH(4, 16) matrices [5]. This particular result motivated the authors to investigate the existence of BH(4, n) matrices in more general and to start enumerating and classifying them for small n. The census of quaternary complex Hadamard matrices up to order 8 were carried out in [14]. The aim of this paper is to continue that work, and give a complete classification of the BH(4, n) matrices up to orders n = 14.This work has two major parts: first, we classify all BH(q, n) matrices using computeraided methods, and secondly, we define parametric families of complex Hadamard matrices by introducing parameters to the BH(q, n) matrices. Parametric families offer a compact way of representing a large number of BH(q, n) matrices, and they also yield information about complex Hadamard matrices.In the previous work [14] the authors collected all BH(4, n) matrices contained in various parametric families from the existing literature and then confirmed with an exhaustive computer search that those are the only examples. Here the situation is exactly the opposite, as it turns out that almost all BH(4, n) matrices of orders n = 12 and 14 found in the current work by the computer search are previously unknown. Therefore we are facing the inverse problem: we need to encode a given collection of BH(4, n) matrices by parametric families.The outline of the paper is as follows. In Section 2 we give some basic definitions and results that are used in later sections. Then in Section 3 we recall the computer-aided classification method of difference matrices from [10] and highlight the main diffe...

Orderly generation of Butson Hadamard matrices

“…Several authors, see e.g. [19,Definition 4.12], [30], consider two BH(n, q) matrices Hadamard equivalent if either can be obtained from the other by performing a finite sequence of monomial equivalence preserving operations, and by replacing every entry by its image under a fixed automorphism of Z q . Given the classification of Butson matrices up to monomial equivalence it is a routine task to determine their number up to Hadamard equivalence.…”

“…The reader might wish to look at the impact of the second column pruning strategy on the number of r × 14 submatrices in Table 1, where we compare the size of the search trees encountered with these two methods during the classification of BH (14,4) We have observed earlier that the computational cost of equivalence testing is independent of the complexity q when orderly generation is used. This is in contrast with the method of canonical augmentation employed earlier in [30] which relies on graph representation of the r × n rectangular orthogonal matrices with qth root entries on 3q(r + n) + r vertices. See [28], [29] for more on graph representation of Butson matrices.…”

“…Remark 3.2. In a prequel to this work [30] we employed the method of canonical augmentation [36, Section 4.2.3] to solve the more general problem of classification of all rectangular orthogonal matrices. Here we solve the relaxed problem of classification of those matrices which can be a constituent of an orderly-generated Butson matrix.…”

Complex Conference Matrices, Complex Hadamard Matrices and Complex Equiangular Tight Frames

“…From Theorem 6 we deduce for instance the existence of (6, 3), (10, 5), (14,7), (18,9), (26, 13)....CET F s. It turns out that for any odd integer k smaller than 50 we may construct a (2k, k) CET F except possibly in the cases k = 11, 17, 23, 29, 33, 35, 39, 43, 47.…”

“…If we put b = ±i then H = C(1, b) ± iI is a quaternary complex Hadamard matrix of order 2k, that is its entries are fourth roots of unity ( [14]). The block construction (1) yields a quaternary Hadamard matrix of order 4k 2 .…”