Table[Det[FAMapMatrix[ToFARule[m,{2,2}],1,2]],{m,0,255}]
{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,-1,0,-1,0,0,1,0,1,-1,0,-1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,-1,1,0,1,0,0,-1,0,-1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,-1,0,-1,0,0,1,0,1,-1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0,-1,0,-1,1,0,1,0,0,-1,0,-1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
Count[%,0]
176
256-%
80
Table[Det[FAMapMatrix[ToFARule[m,{3,2}],1,2]],{m,0,6^6-1}];
Count[%,0]
34992
Table[Det[FAMapMatrix[ToFARule[m,{3,2}],1,3]],{m,0,6^6-1}];
Count[%,0]
37152
Table[Det[FAMapMatrix[ToFARule[m,{3,2}],1,4]],{m,0,6^6-1}];
Count[%,0]
37152
6^6-%
9504
6^6
46656