A core is the lower right $(N-1)\times (N-1)$ submatrix of the dephased form $H'$ of an $N\times N$ complex Hadamard matrix $H$.
For instance, given the matrix $$F_4^{(1)}(a)=\left[\begin{array}{rrrr} 1 & 1 & 1 & 1\\ 1 & i\exp(ia) & -1 & -i\exp(ia)\\ 1 & -1 & 1 & -1\\ 1 & -i\exp(ia) & -1 & i\exp(ia) \end{array}\right]$$ its core consists of $$\left[\begin{array}{rrr} i\exp(ia) & -1 & -i\exp(ia)\\ -1 & 1 & -1\\ -i\exp(ia) & -1 & i\exp(ia) \end{array}\right].$$