Using Hadamard Form of a Matrix in the Block Cipher
This is evident. Let $B$ be a $2^n \times 2^n$ matrix. We have:


  $$a_{i\oplus (j+2^{n-1})}=a_{(2^{n-1}+i)\oplus j},a_{i\oplus j}=a_{(2^{n-1}+i)\oplus (2^{n-1}+j)}.$$


So, $B$ has the following form and therefore is a Hadamard matrix as follows
$$B=
\left(
\begin{array}{cc}
U &amp; V \\
V &amp; U
\end{array}
\right)_{2^n\times 2^n}\, $$