$G_6^{(4)}$

A non-affine four-parameter family of complex Hadamard matrices found by F. Szöllősi.

We start with a submatrix $$E(a,b,c,d)=\left[\begin{array}{ccc} 1 & 1 & 1\\ 1 & a & b\\ 1 & c & d \end{array}\right]$$ and embed it into a complex Hadamard matrix of order $6$ $$G_6^{(4)}(a,b,c,d)=\left[\begin{array}{llllll} 1 & 1 & 1 & 1 & 1 & 1\\ 1 & a & b & e & s_1 & s_2\\ 1 & c & d & f & s_3 & s_4\\ %\hline 1 & g & h & \star & \star & \star\\ 1 & t_1 & t_2 & \star & \star & \star\\ 1 & t_2 & t_4 & \star & \star & \star \end{array}\right]\equiv\left[\begin{array}{cc}E& B\\C&D\end{array}\right]$$ with $3\times 3$ blocks $E$, $B$, $C$ and $D$ in two steps:

  1. construct the submatrices $B$ and $C$ featuring unimodular entries to obtain three orthogonal rows and columns of $G_6$,
  2. find the unique lower right submatrix $D$ to get a unitary matrix $$D=-CE^{\ast}\left(B^{-1}\right)^{\ast}$$

The elements: $e$, $f$, $g$, $h$, $s_k$ and $t_k$ depend on $a$, $b$, $c$ and $d$, although the construction is non-trivial. See [107].

We provide the script in Mathematica by FS that is able to construct "random" generic matrices according to the high level perspective outlined in Section 3 of [107].

Conjecture: The set of complex Hadamard matrices of order $6$ consists of: the isolated matrix $S_6^{(0)}$ and the four-parameter generic family $G_6^{(4)}$.