A family of complex Hadamard matrices generated by the matrix $D_{10}$ was found by F. Szöllősi [25]. It is $$D_{10}^{(3)}(a,b,c)=D_{10}\circ{\rm EXP}\left(i R_{D_{10}^{(3)}}(a,b,c)\right),$$ where
$$D_{10}=\left[\begin{array}{rrrrrrrrrr} 1& 1& 1& 1& 1& 1& 1& 1& 1& 1\\ 1&-1&-i&-i&-i&-i& i& i& i& i\\ 1&-i&-1& i& i&-i&-i&-i& i& i\\ 1&-i& i&-1&-i& i&-i& i&-i& i\\ 1&-i& i&-i&-1& i& i&-i& i&-i\\ 1&-i&-i& i& i&-1& i& i&-i&-i\\ 1& i&-i&-i& i& i&-1&-i&-i& i\\ 1& i&-i& i&-i& i&-i&-1& i&-i\\ 1& i& i&-i& i&-i&-i& i&-1&-i\\ 1& i& i& i&-i&-i& i&-i&-i&-1\end{array}\right]$$ and $$R_{D_{10}^{(3)}}(a,b,c)=\left[\begin{array}{rrrrrrrrrr} \bullet& \bullet& \bullet&\bullet& \bullet&\bullet& \bullet&\bullet& \bullet& \bullet\\ \bullet& \bullet& a-b&a&-c&\bullet&-c&a&a-b& \bullet\\ \bullet& b-a& \bullet&b&-c&\bullet&-c&b& \bullet&b-a\\ \bullet& -a& -b&\bullet& \bullet&\bullet& \bullet&\bullet& -b& -a\\ \bullet& c& c&\bullet& \bullet&\bullet& \bullet&\bullet& c& c\\ \bullet& \bullet& \bullet&\bullet& \bullet&\bullet& \bullet&\bullet& \bullet& \bullet\\ \bullet& c& c&\bullet& \bullet&\bullet& \bullet&\bullet& c& c\\ \bullet& -a& -b&\bullet& \bullet&\bullet& \bullet&\bullet& -b& -a\\ \bullet& b-a& \bullet&b&-c&\bullet&-c&b& \bullet&b-a\\ \bullet& \bullet&a-b&a&-c&\bullet&-c&a&a-b& \bullet \end{array}\right].$$The defect $d\left(D_{10}^{(3)}\right)=16$, so it is not known whether this family is maximal affine, or further parameters can be introduced.