Addition modulo n can be handled by inserting Modulus  n in NullSpace in the previous note.

A way to find candidates for q is to compute NullSpace[Table[With[{u = Table[Random[Integer, {0, k - 1}], {m}]}, BC[CAStep[u]] - BC[u]], {s}]] for progressively larger m and s , and to see what lists continue to appear.