A matrix \(C\) of order \(n\) is called a conference matrix if:

1.

Its main diagonal consists of zeros;

2.

Its entries that are not lying in the main diagonal have modulus \(1\);

3.

\(C^*C=(n-1)I\).

Two complex conference matrices \(C\), \(\tilde C\) are said to be equivalent if there exist unitary diagonal matrices \(D_1\), \(D_2\) and a permutation matrix \(S\) such that \(\tilde C=D_1SCS^TD_2\).
The authors define two types of conference matrices for which consecutive rows are cyclic permutations of the first one. Using these two as some sort of base they discuss the classification of complex conference matrices of small orders.
First, they prove that every complex conference matrix of order \(4\) is equivalent to the (unique) real skew-symmetric conference matrix of order \(4\): \[ \begin{pmatrix} 0&1&1&1\\ -1&0&1&-1\\ -1&-1&0&1\\ -1&1&-1&0\\ \end{pmatrix}. \] Next, they show that every complex conference matrix of order \(5\) is equivalent to \[ \begin{pmatrix} 0&1&\omega&\omega&1\\ 1&0&1&\omega&\omega\\ \omega&1&0&1&\omega\\ \omega&\omega&1&0&1\\ 1&\omega&\omega&1&0\\ \end{pmatrix}, \] where \(\omega\) is a root of \(1\) of degree \(3\).
The case when \(n=6\) is more complicated. Therefore, the authors describe symmetric, skew-symmetric and Hermitian conference matrices of order \(6\).
The obtained results are used for the construction of equi-isoclinic planes in Euclidean spaces, in particular answering to a question posed in [P. W. H. Lemmens and J. J. Seidel, Nederl. Akad. Wet., Proc., Ser. A 76, 98–107 (1973; Zbl 0272.50008)].