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

Introduction

Preliminaries

Let $H$ be a complex Hadamard matrix of order $n$ and $2\leq k\leq n-2$ be an integral number. Then the number of $k\times k$ vanishing minors of $H$ is invariant, up to equivalence.

The number of $k\times k$ vanishing minors is just a special case of a more powerful invariant, the fingerprint , but it is sufficient for our purposes.

Constructing Butson-type Hadamard matrices

The computer-aided methods employed in the classification of Butson-type Hadamard matrices in this work are very similar to the methods used for the classification of difference matrices over cyclic groups . Therefore, we give here only a summary of the relevant ideas and describe in detail only the points where this work differs from the work done on difference matrices.

In the tree each $X$ occurs only once but there are typically many nodes $Y$ with $X\cong Y$. Such a node $Y$ has also a sequence of ancestors from which it has been constructed:

Even though $X\cong Y$ the ancestor sequences (1) and (2) need not consist of the same nodes up to equivalence. This means that the sequences (1) and (2) can be distinct on the level of equivalence classes even if $X\cong Y$. The main idea is now to exploit such differences among equivalent nodes in rejecting equivalent nodes.

Let $T_{nr}$ denote the set of all non-root nodes in the search tree $T$. Associate with every object $X\in T_{nr}$ a weak canonical parent $w(X)\in T$ such that the following property holds:

The function $w$ defines for every non-root node $X$ a sequence of objects analogous to (1):

Because of (3), any two equivalent objects have identical sequences (4) on the level of equivalence classes of matrices. When the search tree is traversed in depth-first order, a node $X$ and the subtree rooted at it is considered only if it has been constructed in the canonical way specified by (4); that is, every matrix in the ancestor sequence (1) should be equivalent to the matrix in the corresponding position in the sequence (4). By (3) we obtain

In other words, equivalent matrices generated by weak canonical augmentation have equivalent parent matrices, and assuming that the same holds for the parents, this implies that equivalent matrices must be siblings in the search tree. This reduces the size of the search tree dramatically.

Transforming a matrix to a graph yields also a weak canonical function $w$ having the property (3). Let $P(X)$ be the set of matrices obtained by removing a row from a matrix $X\in T_{nr}$ and let $\leq_{g}$ denote a total order on the set of canonical graphs. We define $w(X)$ as the matrix $Y\in P(X)$ which has the smallest canonical graph under $\leq_{g}$ among the canonical graphs of matrices in $P(X)$.

Parametrizing complex Hadamard matrices

Let $H$ be a dephased complex Hadamard matrix of order $n$ and suppose that there exist a pair of rows in $H$, say $u$ and $v$, such that for every $i=1,2,\ldots,n$, $u_{i}^{2}=v_{i}^{2}$. Then for all such $i$ for which $u_{i}+v_{i}=0$ replace $u_{i}$ with $\alpha u_{i}$ and $v_{i}$ with $\alpha v_{i}$, where $\alpha$ is a unimodular complex number to obtain a one-parameter family of complex Hadamard matrices $H(\alpha)$.

Let $H$ be a dephased complex Hadamard matrix with the following block structure

where $a$ and $b$ are arbitrary unimodular numbers. Then, after replacing the row vectors $y$ with $\alpha y$ and $w$ with $\overline{\alpha}w$ we obtain a one-parameter family of complex Hadamard matrices $H(\alpha)$ where $\alpha$ is unimodular. If, in addition, $b=a$ we can continue by replacing $w$ in $H(\alpha)$ with $\alpha\beta w$ to obtain a two-parameter family of complex Hadamard matrices $H(\alpha,\beta)$ where $\alpha$ and $\beta$ are unimodular.

We need to show that the rows of $H(\alpha,\beta)$ are pairwise orthogonal. From the orthogonality of the first three rows of $H$ we get

and hence the first three rows of $H(\alpha,\beta)$ are pairwise orthogonal. Similarly, it is easily seen that the rest of the rows (beyond the first three) are pairwise orthogonal within themselves. Additionally, the first row is trivially orthogonal to all further rows. Therefore it remains to be seen that the second and third rows of $H(\alpha,\beta)$ are orthogonal to all rows below them.

We show first that they are orthogonal to the rows which are of type $[1,z_{i},z_{i},A_{i},B_{i}]$, $i=1,\ldots,r$. In the original matrix $H$ (i.e., prior to parametrizing) we have

and hence $\left\langle B_{i},y\right\rangle=0$ for every $i=1,\ldots,r$. It follows that after parametrization equations (5) and (6) remain valid.

We proceed by proving that rows that are of type $[1,w_{i},-w_{i},C_{i},D_{i}]$, $i=1,\ldots,s$, after parametrization, are orthogonal to the second and third row of $H(\alpha,\beta)$. Again, in the original matrix $H$ we have

and hence $\left\langle C_{i},x\right\rangle=-1$ for every $i=1,\ldots,s$. It follows, that (7) are (8) are valid, provided that

holds and therefore (7) and (8) remains true, after parametrization. If, in addition, $b=a$, then $\left\langle D_{i},y\right\rangle=0$ for every $i=1,\ldots,s$, and hence (9) holds, independently of the scalar factor in $w$.

The one-parameter family $H(\alpha)$ can be considered as $H(\alpha,\overline{\alpha})$. The equations (5) - (8) hold as the condition $a=b$ is not required for them. From the original matrix $H$ we get

and (9) holds for $H(\alpha,\overline{\alpha})$ also when $a\neq b$. ∎

If $a$ and $b$ are as in Theorem 4.2, then it is easy to see that the real part of $a\overline{b}$ is an integral number, and therefore $b\in\pm a\cdot\{1,\omega,\omega^{2},\mathbf{i}\}$, where $\omega$ is a primitive complex third root of unity. Therefore one hopes to apply the parametrizing scheme described for complex Hadamard matrices with fourth and/or sixth roots of unity.

Theorem 4.2 describes a local property of the complex Hadamard matrix $H$. Its conditions can be fairly easily checked, even by hand, and it can be implemented as a computer program to construct infinite families automatically.

which, after linearizing the exponential function (i.e., replacing it with its first order Taylor expansion) leads to (10). Therefore those phasing matrices $R$ satisfying the real linear system (10) lead to parametric families of complex Hadamard matrices in a neighborhood of the initial matrix $H$, up to first order; however, (11) is far more restrictive further decreasing the number of parameters in $R$ in general.

Note that the degree of freedom $m$ in the defect of an $n\times n$ matrix is decreased by $2n-1$ as this many parameters can always be introduced into a complex Hadamard matrix via multiplication by unitary diagonal matrices. This operation, however, does not yield new complex Hadamard matrices, up to equivalence, and therefore only dephased families are considered. Matrices that cannot be parametrized in any other way are called isolated. The most important properties of the defect are summarized in the following result from .

Let $H$ be a complex Hadamard matrix. Then

Note that part (c) does not require the smoothness condition and as a result it does not follow from part (b).

The quaternary complex Hadamard matrices of order 101010

The two remaining matrices can be obtained from complex Golay sequences , and they belong to the family

The quaternary complex Hadamard matrices of order 121212

The quaternary complex Hadamard matrices of order 141414

The exhaustive computer search yielded the following result:

References