Paull matrix configurations

Question

In this paull matrix creation, instead of {{0,1,2},{1,2,0},{2,0,1}} can I write the other combination {{0,1,2},{2,0,1},{1,2,0}} in that case network will be different. or if not can you please explain how I can construct that paull matrix from the phi? 

Answer 1

The Paull matrix given in the solution is as follows

    0 1 2
    1 2 0
    2 0 1

We can easily check that this solution is correct, since each one of the indices 0,1,2 appears exactly once in every row and column. The same is the case with your suggestion

    0 1 2
    2 0 1
    1 2 0

This Paull matrix is also correct, since every middle switch with index 0,1,2 has a unique left connection and a unique right connection since each one of the indices 0,1,2 appears exactly once in every row and column.

Clearly, the connections will be routed differently with the two matrices, but in both cases, the given permutation can be correctly established.