Template:Zhegalkin matrix/matrix pi xor phi



Let $$\pi = \Pi_2$$ (length 16) , $$\phi = \Phi_3$$ (length 16) and $$\xi = \Xi_3$$ (length 256). Let $$I_k =  \{ i \mid \xi_i = \phi_k \}$$ be the set of places where $$\xi$$ has the entry $$\phi_k$$.

$$\pi$$ is the left column of the following matrix. $$\phi$$ is its top row, and also shown in the column to its right. The matrix entries are the bitwise XORs of its left column and top row. $$I_k$$ is row $$k$$ as a set of numbers: $$\{ \pi_k \oplus f \mid f \in \phi \}$$

Example:  In which places is $$\xi$$ equal to $$\phi_{10} = 168$$? $$\pi_{10} = 2 ~ \implies ~ I_{10} =  \{ 2 \oplus f \mid f \in \phi \}  =  \{2, 28, ... , 226, 252 \}$$ It can be easily seen, that the first and the last entry 168 in $$\xi$$ is in places 2 and 252.