The only maximal affine Hadamard family stemming from the Fourier matrix $$F_{16} = \left[\begin{array}{llllllllllllllll} 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1\\ 1& w& w^2& w^3& w^4& w^5& w^6& w^7& w^8& w^9& w^{10} & w^{11}& w^{12}& w^{13}& w^{14}& w^{15}\\ 1& w^2& w^4& w^6& w^8& w^{10}& w^{12}& w^{14}& 1& w^2& w^4& w^6& w^8& w^{10}& w^{12}& w^{14}\\ 1& w^3& w^6& w^9& w^{12}& w^{15}& w^2& w^5& w^8& w^{11}& w^{14}& w& w^4& w^7& w^{10}& w^{13}\\ 1& w^4& w^8& w^{12}& 1& w^4& w^8& w^{12}& 1& w^4& w^8& w^{12}& 1& w^4& w^8& w^{12}\\ 1& w^5& w^{10}& w^{15}& w^4& w^9& w^{14}& w^3& w^8& w^{13}& w^2& w^7& w^{12}& w& w^6& w^{11}\\ 1& w^6& w^{12}& w^2& w^8& w^{14}& w^4& w^{10}& 1& w^6& w^{12}& w^2& w^8& w^{14}& w^4& w^{10}\\ 1& w^7& w^{14}& w^5& w^{12}& w^3& w^{10}& w& w^8& w^{15}& w^6& w^{13}& w^4& w^{11}& w^2& w^9\\ 1& w^8& 1& w^8& 1& w^8& 1& w^8& 1& w^8& 1& w^8& 1& w^8& 1& w^8\\ 1& w^9& w^2& w^{11}& w^4& w^{13}& w^6& w^{15}& w^8& w& w^{10}& w^3& w^{12}& w^5& w^{14}& w^7\\ 1& w^{10}& w^4& w^{14}& w^8& w^2& w^{12}& w^6& 1& w^{10}& w^4& w^{14}& w^8& w^2& w^{12}& w^6\\ 1& w^{11}& w^6& w& w^{12}& w^7& w^2& w^{13}& w^8& w^3& w^{14}& w^9& w^4& w^{15}& w^{10}& w^5\\ 1& w^{12}& w^8& w^4& 1& w^{12}& w^8& w^4& 1& w^{12}& w^8& w^4& 1& w^{12}& w^8& w^4\\ 1& w^{13}& w^{10}& w^7& w^4& w& w^{14}& w^{11}& w^8& w^5& w^2& w^{15}& w^{12}& w^9& w^6& w^3\\ 1& w^{14}& w^{12}& w^{10}& w^8& w^6& w^4& w^2& 1& w^{14}& w^{12}& w^{10}& w^8& w^6& w^4& w^2\\ 1& w^{15}& w^{14}& w^{13}& w^{12}& w^{11}& w^{10}& w^9& w^8& w^7& w^6& w^5& w^4& w^3& w^2& w \end{array}\right] : \quad w=\exp\frac{2i\pi}{16}$$ is the $17$-parameter self-cognate family in the form $$F_{16}^{(17)}(a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,r)=F_{16}\circ{\rm EXP}\left(i R_{F_{16}^{(17)}}(a,...,r)\right),$$ where the core of matrix of phases is \begin{align*} &{\rm core}\left(R_{F_{16}^{(17)}}(a,...,r)\right)\\ &= \left[\begin{array}{rrrrrrrrrrrrrrr} a& b& c& d& e& f& g& \bullet & a& b& c& d& e& f& g \\ h& i& j& \bullet & h& i& j& \bullet & h& i& j& \bullet & h& i& j\\ k& l& m& d& e-a+k& f-b+l& g-c+m& \bullet & k& l& m& d& e-a+k& f-b+l& g-c+m \\ n& \bullet & n& \bullet & n& \bullet & n& \bullet & n& \bullet & n& \bullet & n& \bullet & n \\ o& b& c-a+o& d& e-a+o& f& g-a+o& \bullet & o& b& c-a+o& d& e-a+o& f& g-a+o \\ p& i& j-h+p& \bullet & p& i& j-h+p& \bullet & p& i& j-h+p& \bullet & p& i& j-h+p\\ r& l& m-k+r& d& e-a+r& f-b+l& g-c-k+m+r& \bullet & r& l& m-k+r& d& e-a+r& f-b+l& g-c-k+m+r \\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet \\ a& b& c& d& e& f& g& \bullet & a& b& c& d& e& f& g \\ h& i& j& \bullet & h& i& j& \bullet & h& i& j& \bullet & h& i& j\\ k& l& m& d& e-a+k& f-b+l& g-c+m& \bullet & k& l& m& d& e-a+k& f-b+l& g-c+m \\ n& \bullet & n& \bullet & n& \bullet & n& \bullet & n& \bullet & n& \bullet & n& \bullet & n \\ o& b& c-a+o& d& e-a+o& f& g-a+o& \bullet & o& b& c-a+o& d& e-a+o& f& g-a+o \\ p& i& j-h+p& \bullet & p& i& j-h+p& \bullet & p& i& j-h+p& \bullet & p& i& j-h+p\\ r& l& m-k+r& d& e-a+r& f-b+l& g-c-k+m+r& \bullet & r& l& m-k+r& d& e-a+r& f-b+l& g-c-k+m+r \end{array}\right]. \end{align*}