AUSTRALASIAN JOURNAL OF COMBINATORICS
Volume 55 (2013), Pages 31–34

A note on the existence of BH(19, 6) matrices
Ferenc Szöllősi
Department of Mathematics and its Applications
Central European University
H-1051, Nádor u. 9, Budapest
Hungaryszoferi@gmail.com
Dedicated to Professor Kathy J. Horadam on the occasion of her 60th birthday
Abstract
In this note we utilize a non-trivial block approach due to M. Petrescu toexhibit a Butson-type complex Hadamard matrix of order 19, composedof sixth roots of unity.

1A new Butson-type complex Hadamard matrix

A complex Hadamard matrix H is an n × n complex matrix with unimodular entries such that HH ∗ = nIn , where ∗ denotes the Hermitian adjoint and In is theidentity matrix of order n. Throughout this note we are concerned with the special case when the entries are some qth roots of unity. These matrices are called
Butson-type complex Hadamard matrices and are denoted by BH(n, q) [1]. The existence of BH(n, q) matrices is wide open in general, but it is believed that real, i.e.
BH(4k, 2) matrices exist for every integer number k. This is the famous Hadamard
conjecture. Butson-type complex Hadamard matrices have applications in signal
processing, coding theory [4] and harmonic analysis [6], among other things. They
can also lead to constructions of real Hadamard matrices. This approach was demonstrated very recently in [3], where BH(n, 6) matrices were considered. For further
results on BH(n, q) matrices we refer the reader to Horadam’s celebrated book [5].
The main result of this note is the following
Theorem 1.1. There exists a BH(19, 6) matrix.
Order n = 19 was listed as the smallest outstanding order of BH(n, 6) matricesin [2]. Addressing the existence of complex Hadamard matrices of prime orders
is a notoriously diﬃcult problem. One of the reasons for this is that the standard
construction methods developed for the study of real Hadamard matrices and various¨˝
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OSI32

orthogonal arrays fundamentally rely on “plug-in” methods and block constructions
resulting in objects of composite order.
However, a non-standard block approach was utilized by Petrescu in his PhD
thesis [7], who exhibited various complex Hadamard matrices of prime orders. We
recall his method as follows. Let s be a positive integer, X, Y be s × s, D be an(s + 1) × (s + 1) matrix, and assume that the s × (s + 1) matrix T is composed ofs noninitial rows of a normalized complex Hadamard matrix of order s + 1. Thefollowing matrix of order 3s + 1⎡⎤
X Y TH=⎣ Y X T ⎦T∗ T∗ Dis called Petrescu’s array. The matrix H is complex Hadamard if and only if theentries of X, Y, D and T are unimodular and the following orthogonality equations
are satisﬁed:
2T ∗ T + DD∗
XX + Y Y ∗ + T T ∗(X + Y )T + T D∗
XY ∗ + Y X ∗ + T T ∗
∗

==
==

(3s + 1)Is+1 ,(3s + 1)Is ,0,0.

Let us denote by Js+1 the all-1 matrix of order s + 1. The system of equations above
can be transformed into the following equivalent, but more informative one (see [7]for details):
DD∗ = D∗ D = (s − 1)Is+1 + 2Js+1 ,DJ = JD,1
X +Y = −T D∗T ∗,s+1(X − Y )(X − Y )∗ = (3s + 1)Is .

(1)
(2)
(3)
(4)

Thus the lower right corner D is a normal, regular matrix, and hence in view of (1)can be thought as some kind of generalized, “complex biplane”. The sum X + Y isessentially a unitary transform of D∗ , while the diﬀerence X − Y is an orthogonalmatrix.
In order to prove Theorem 1.1 we take s = 6 and look for some solutions of theabove equations in which all of the matrix entries are some 6th roots of unity. First
we search for candidate matrices D satisfying (1) and (2), and then we search for asuitable, normalized BH(7, 6) matrix, whose six noninitial rows will constitute T suchthat the right hand side of (3) is a matrix in which all entries are sums of some sixth
roots of unity. Finally, we decompose the right hand side of (3) into some matrices
X and Y and check whether they meet the ﬁnal condition (4). After implementing
this rather straightforward algorithm in Mathematica, we have found the followingEXISTENCE OF BH(19, 6) MATRICES

matrix, virtually in seconds:⎡
3 0 1 1 0
⎢ 0 0 1 3 3⎢⎢ 0 0 1 4 2⎢⎢ 1 2 4 2 1⎢⎢ 2 5 4 3 2⎢⎢ 0 3 5 4 5⎢⎢ 5 4 3 5 3⎢⎢ 4 2 4 5 1⎢⎢ 2 4 3 2 4⎢
W19 = ⎢
⎢ 4 4 2 4 5⎢ 4 2 0 1 4⎢⎢ 4 5 3 1 3⎢⎢ 5 5 3 3 2⎢⎢ 5 2 3 1 5⎢⎢ 3 3 5 5 1⎢⎢ 1 3 2 5 3⎢⎢ 2 5 1 3 5⎢⎣ 3 1 5 2 30 0 0 0 00142002510241325350542444300120553123042442500025352335103432031114453352150

5524111342343
15532031454303212525135302510240142001325350113345113345001011414351314351300310013311543311540
10041133534131534131030
1
100415313415313
1000
1413514313514
3
111
1
4000

⎤
00 ⎥
⎥
0 ⎥
⎥
0 ⎥
⎥
0 ⎥
⎥
0 ⎥
⎥
0 ⎥
⎥
0 ⎥
⎥
0 ⎥
⎥
0 ⎥
⎥.
0 ⎥
⎥
0 ⎥
⎥
3 ⎥
⎥
0 ⎥
⎥
0 ⎥
⎥
1 ⎥
⎥
0 ⎥
⎥
1 ⎦
02πi

In the matrix W19 above we have replaced the sixth roots of unity e 6 j with j,
j = 0, 1, . . . , 5 for typographical reasons. A more in-depth analysis of this approach
along with a handful of additional examples of Petrescu-type complex Hadamard
matrices will be presented elsewhere.

Acknowledgements
The results of this note were obtained during a research visit to Professor Robert
Craigen. The author acknowledges the warm hospitality of the Department of Mathematics at University of Manitoba. This work was supported by the Hungarian National Research Fund OTKA K-77748 and by the Doctoral Research Support Grantof CEU.

References[1] A. T. Butson, Generalized Hadamard matrices, Proc. Amer. Math. Soc. 13(1962), 894–898.
[2] C. J. Colbourn and J. Dinitz, eds., Chapter V., The CRC Handbook ofCombinatorial Designs, CRC Press, Boca Raton, (1996).
[3] B. Compton, R. Craigen and W. de Launey, Unreal BH(n, 6)’s andHadamard matrices, Preprint, (2008).

34¨˝
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OSI[4] M. Harada, C. Lam and V. D. Tonchev, Symmetric (4, 4)-nets and Generalized Hadamard Matrices Over Groups of Order 4, Des., Codes Crypto. 34(2005), 71–87.
[5] K. J. Horadam, Hadamard Matrices and Their Applications, Princeton University Press, (2007).
[6] M. Kolountzakis and M. Matolcsi, Complex Hadamard matrices and thespectral set conjecture, Collect. Math. Vol. Extra (2006), 281–291.
[7] M. Petrescu, Existence of continuous families of complex Hadamard matricesof certain prime dimensions, PhD thesis, UCLA, (1997).

(Received 21 Dec 2011)