(* {3,8},{2,4,9,16}; large lists are at the end *)

Dynamic[{j,i,g, mark}]
Clear[n,B,R,Tile,A,U,mark,T,g,x];

n=144;

L=Expand[Cyclotomic[8,x^9]Cyclotomic[3,x^(16)]];

B=Array[0&,Length[L]];
For[j=2, j≤Length[L], B[[j]]=Mod[L[[j,2]],n];
j++];
B⋃{}

Tile=Array[0&,1000000];
R=Array[0&,n];
A=Array[200&,{n/Length[B],n,Length[B]}];
U=Array[0&,n/Length[B]];
U[[1]]=B;
mark=Array[1&,n/Length[B]+1];
T=Array[{0}&,n/Length[B]];
g=1;

For[i=1, i>0,
If[Length[U[[i]]]==n,  Tile[[g]]=T[[i]]⋃{};g=g+1;T[[i]]=T[[i-1]];mark[[i]]=1;A[[i]]=Array[200&,{n,Length[B]}];i=i-1,
For[j=1, j≤n,k=100;
If[Not[MemberQ[U[[i]],Mod[j,n]]], k=0;
For[m=1, m≤Length[B],
If[U[[i]]⋂ Mod[(j-B[[m]])+B,n]=={}, k=k+1; A[[i,j,m]]=Mod[j-B[[m]],n]];
m++]];
R[[j]]=k;
j++];
min=Min[R]; 
If[(Not[min==0]),
t=Catch[Do[If[R[[ii]]==min, Throw[ii]],{ii,n}]];
If[mark[[i]]≤Length[B],
For[b=mark[[i]], b<=Length[B],mark[[i]]=b+1;z=0;
If[Not[A[[i,t,b]]==200], U[[i+1]]=U[[i]]⋃Mod[A[[i,t,b]]+B,n]; T[[i+1]]=T[[i]]⋃{A[[i,t,b]]}; b=Length[B]+1;z=1;i=i+1;];
b++];
If[z==0, mark[[i]]=1;A[[i]]=Array[200&,{n,Length[B]}];i=i-1];
, mark[[i]]=1;A[[i]]=Array[200&,{n,Length[B]}];T[[i]]=T[[i-1]];i=i-1];
, mark[[i]]=1;A[[i]]=Array[200&,{n,Length[B]}];T[[i]]=T[[i-1]];i=i-1];
];
];

Tile=Drop[Tile⋃{},1];
Length[Tile]
Tile;

H=Array[{0}&,Length[Tile[[1]]]+1];
K=Tile[[3]];
For[j=1, j≤Length[K], 
H[[j+1]]=Mod[(K[[j]]+B)⋃ H[[j]],n];
j++];
H[[Length[K]]]⋃{} 

For[j=1, j≤Length[Tile], P=Tile[[j]];
For[i=2, i<=Length[P], 
If[MemberQ[Tile,Mod[P-P[[i]],n]⋃{}], Tile[[j]]=0;i=Length[P]+1];
i++];
j++]; 
Tile=Drop[Tile⋃{},1];
Length[Tile]
Tile

(*{{3,8},{2,4,9,16}} here come the large lists*)

Dynamic[{j,i,g, mark}]
Clear[n,B,R,Tile,A,U,mark,T,g,x];

n=144;


B={0,9,17,26,27,34,39,40,48,51,57,65,74,82,88,96,99,105,111,113,122,123,130,136};


Tile=Array[0&,1000000];
R=Array[0&,n];
A=Array[200&,{n/Length[B],n,Length[B]}];
U=Array[0&,n/Length[B]];
U[[1]]=B;
mark=Array[1&,n/Length[B]+1];
T=Array[{0}&,n/Length[B]];
g=1;

For[i=1, i>0,
If[Length[U[[i]]]==n,  Tile[[g]]=T[[i]]⋃{};g=g+1;T[[i]]=T[[i-1]];mark[[i]]=1;A[[i]]=Array[200&,{n,Length[B]}];i=i-1,
For[j=1, j≤n,k=100;
If[Not[MemberQ[U[[i]],Mod[j,n]]], k=0;
For[m=1, m≤Length[B],
If[U[[i]]⋂ Mod[(j-B[[m]])+B,n]=={}, k=k+1; A[[i,j,m]]=Mod[j-B[[m]],n]];
m++]];
R[[j]]=k;
j++];
min=Min[R]; 
If[(Not[min==0]),
t=Catch[Do[If[R[[ii]]==min, Throw[ii]],{ii,n}]];
If[mark[[i]]≤Length[B],
For[b=mark[[i]], b<=Length[B],mark[[i]]=b+1;z=0;
If[Not[A[[i,t,b]]==200], U[[i+1]]=U[[i]]⋃Mod[A[[i,t,b]]+B,n]; T[[i+1]]=T[[i]]⋃{A[[i,t,b]]}; b=Length[B]+1;z=1;i=i+1;];
b++];
If[z==0, mark[[i]]=1;A[[i]]=Array[200&,{n,Length[B]}];i=i-1];
, mark[[i]]=1;A[[i]]=Array[200&,{n,Length[B]}];T[[i]]=T[[i-1]];i=i-1];
, mark[[i]]=1;A[[i]]=Array[200&,{n,Length[B]}];T[[i]]=T[[i-1]];i=i-1];
];
];

Tile=Drop[Tile⋃{},1];
Length[Tile]
Tile;

H=Array[{0}&,Length[Tile[[1]]]+1];
K=Tile[[3]];
For[j=1, j≤Length[K], 
H[[j+1]]=Mod[(K[[j]]+B)⋃ H[[j]],n];
j++];
H[[Length[K]+1]]⋃{} 

For[j=1, j≤Length[Tile], P=Tile[[j]];
For[i=2, i<=Length[P], 
If[MemberQ[Tile,Mod[P-P[[i]],n]⋃{}], Tile[[j]]=0;i=Length[P]+1];
i++];
j++]; 
Tile=Drop[Tile⋃{},1];
Length[Tile]
Tile

(*{{3,8},{2,4,9,16}}  *)


Dynamic[{t1,t2, dd}]
T1={{0,36,64,80,100,116},{0,44,76,80,108,112},{0,76,92,108,112,128}};



T2={{0,9,17,26,27,34,39,40,48,51,57,65,74,82,88,96,99,105,111,113,122,123,130,136},{0,9,17,26,31,34,40,43,48,55,57,65,74,82,88,96,103,105,113,115,122,127,130,136},{0,9,17,26,34,35,40,47,48,57,59,65,74,82,88,96,105,107,113,119,122,130,131,136},{0,9,17,26,34,39,40,48,51,57,63,65,74,82,88,96,105,111,113,122,123,130,135,136},{0,9,17,26,34,40,43,48,55,57,65,67,74,82,88,96,105,113,115,122,127,130,136,139},{0,9,17,26,34,40,47,48,57,59,65,71,74,82,88,96,105,113,119,122,130,131,136,143},{0,9,18,26,27,33,39,40,48,51,57,66,74,81,88,96,99,105,111,114,122,123,129,136},{0,9,18,26,31,33,40,43,48,55,57,66,74,81,88,96,103,105,114,115,122,127,129,136},{0,9,18,26,33,35,40,47,48,57,59,66,74,81,88,96,105,107,114,119,122,129,131,136},{0,9,18,26,33,39,40,48,51,57,63,66,74,81,88,96,105,111,114,122,123,129,135,136},{0,9,18,26,33,40,43,48,55,57,66,67,74,81,88,96,105,114,115,122,127,129,136,139},{0,9,18,26,33,40,47,48,57,59,66,71,74,81,88,96,105,114,119,122,129,131,136,143},{0,10,17,22,25,31,40,43,48,55,65,70,73,82,88,94,96,103,113,115,121,127,136,142},{0,10,17,22,25,34,40,43,48,55,65,67,73,82,88,94,96,106,113,115,121,127,136,139},{0,10,17,25,27,34,39,40,48,51,58,65,73,82,88,96,99,106,111,113,121,123,130,136},{0,10,17,25,31,34,40,43,48,55,58,65,73,82,88,96,103,106,113,115,121,127,130,136},{0,10,17,25,31,40,43,48,55,58,65,70,73,82,88,96,103,113,115,121,127,130,136,142},{0,10,17,25,34,35,40,47,48,58,59,65,73,82,88,96,106,107,113,119,121,130,131,136},{0,10,17,25,34,39,40,48,51,58,63,65,73,82,88,96,106,111,113,121,123,130,135,136},{0,10,17,25,34,40,43,48,55,58,65,67,73,82,88,96,106,113,115,121,127,130,136,139},{0,10,17,25,34,40,47,48,58,59,65,71,73,82,88,96,106,113,119,121,130,131,136,143},{0,10,18,23,25,31,37,40,48,49,58,66,71,79,88,96,97,106,109,114,119,121,127,136},{0,10,18,23,29,31,40,41,48,53,58,66,71,79,88,96,101,106,113,114,119,125,127,136},{0,10,18,23,31,33,40,45,48,57,58,66,71,79,88,96,105,106,114,117,119,127,129,136},{0,10,18,23,31,37,40,48,49,58,61,66,71,79,88,96,106,109,114,119,121,127,133,136},{0,10,18,23,31,40,41,48,53,58,65,66,71,79,88,96,106,113,114,119,125,127,136,137},{0,10,18,23,31,40,45,48,57,58,66,69,71,79,88,96,106,114,117,119,127,129,136,141},{0,10,18,24,25,31,37,39,48,49,58,66,72,79,87,96,97,106,109,114,120,121,127,135},{0,10,18,24,27,33,39,41,48,51,58,66,72,81,89,96,99,106,111,114,120,123,129,137},{0,10,18,24,29,31,39,41,48,53,58,66,72,79,87,96,101,106,113,114,120,125,127,135},{0,10,18,24,31,33,39,45,48,57,58,66,72,79,87,96,105,106,114,117,120,127,129,135},{0,10,18,24,31,33,41,43,48,55,58,66,72,81,89,96,103,106,114,115,120,127,129,137},{0,10,18,24,31,37,39,48,49,58,61,66,72,79,87,96,106,109,114,120,121,127,133,135},{0,10,18,24,31,39,41,48,53,58,65,66,72,79,87,96,106,113,114,120,125,127,135,137},{0,10,18,24,31,39,45,48,57,58,66,69,72,79,87,96,106,114,117,120,127,129,135,141},{0,10,18,24,33,35,41,47,48,58,59,66,72,81,89,96,106,107,114,119,120,129,131,137},{0,10,18,24,33,39,41,48,51,58,63,66,72,81,89,96,106,111,114,120,123,129,135,137},{0,10,18,24,33,41,43,48,55,58,66,67,72,81,89,96,106,114,115,120,127,129,137,139},{0,10,18,24,33,41,47,48,58,59,66,71,72,81,89,96,106,114,119,120,129,131,137,143},{0,10,18,25,27,33,39,40,48,51,58,66,73,81,88,96,99,106,111,114,121,123,129,136},{0,10,18,25,27,35,37,40,48,49,58,66,75,83,88,96,97,106,109,114,121,123,131,136},{0,10,18,25,31,33,40,43,48,55,58,66,73,81,88,96,103,106,114,115,121,127,129,136},{0,10,18,25,33,35,40,47,48,58,59,66,73,81,88,96,106,107,114,119,121,129,131,136},{0,10,18,25,33,39,40,48,51,58,63,66,73,81,88,96,106,111,114,121,123,129,135,136},{0,10,18,25,33,40,43,48,55,58,66,67,73,81,88,96,106,114,115,121,127,129,136,139},{0,10,18,25,33,40,47,48,58,59,66,71,73,81,88,96,106,114,119,121,129,131,136,143},{0,10,18,27,29,35,40,41,48,53,58,66,75,83,88,96,101,106,113,114,123,125,131,136},{0,10,18,27,33,35,40,45,48,57,58,66,75,83,88,96,105,106,114,117,123,129,131,136},{0,10,18,27,33,39,41,48,51,58,60,66,72,81,89,99,106,111,114,120,123,129,132,137},{0,10,18,27,35,37,40,48,49,58,61,66,75,83,88,96,106,109,114,121,123,131,133,136},{0,10,18,27,35,40,41,48,53,58,65,66,75,83,88,96,106,113,114,123,125,131,136,137},{0,10,18,27,35,40,45,48,57,58,66,69,75,83,88,96,106,114,117,123,129,131,136,141},{0,10,19,22,27,33,34,40,45,48,57,67,75,82,88,94,96,105,106,115,117,123,129,136},{0,10,19,22,27,33,40,45,48,57,67,70,75,82,88,94,96,105,115,117,123,129,136,142},{0,10,19,22,27,34,40,45,48,57,67,69,75,82,88,94,96,106,115,117,123,129,136,141},{0,10,19,25,27,34,37,40,48,49,58,67,75,82,88,96,97,106,109,115,121,123,130,136},{0,10,19,27,29,34,40,41,48,53,58,67,75,82,88,96,101,106,113,115,123,125,130,136},{0,10,19,27,33,34,40,45,48,57,58,67,75,82,88,96,105,106,115,117,123,129,130,136},{0,10,19,27,33,40,45,48,57,58,67,70,75,82,88,96,105,115,117,123,129,130,136,142},{0,10,19,27,34,37,40,48,49,58,61,67,75,82,88,96,106,109,115,121,123,130,133,136},{0,10,19,27,34,40,41,48,53,58,65,67,75,82,88,96,106,113,115,123,125,130,136,137},{0,10,19,27,34,40,45,48,57,58,67,69,75,82,88,96,106,115,117,123,129,130,136,141},{0,11,12,19,24,25,34,37,42,49,59,67,72,82,84,90,96,97,107,109,115,121,130,138},{0,11,12,19,24,26,33,38,41,50,59,67,72,81,84,89,96,98,107,110,115,122,129,137},{0,11,12,19,25,34,37,42,49,59,60,67,72,82,84,90,97,107,109,115,121,130,132,138},{0,11,12,19,26,33,38,41,50,59,60,67,72,81,84,89,98,107,110,115,122,129,132,137},{0,11,13,19,25,34,37,42,48,59,60,67,72,82,85,90,97,107,109,115,120,130,132,138},{0,11,14,19,26,33,38,41,48,59,60,67,72,81,86,89,98,107,110,115,120,129,132,137},{0,11,17,23,25,30,40,42,48,54,65,71,73,83,88,95,96,102,113,114,121,126,136,143},{0,11,17,23,25,34,35,40,46,48,58,65,73,83,88,95,96,106,107,113,118,121,130,136},{0,11,17,23,25,34,40,46,48,58,65,71,73,83,88,95,96,106,113,118,121,130,136,143},{0,11,17,23,25,35,40,42,48,54,65,66,73,83,88,95,96,107,113,114,121,126,136,138},{0,11,17,23,25,35,40,46,48,58,65,70,73,83,88,95,96,107,113,118,121,130,136,142},{0,11,17,25,26,35,38,40,48,50,59,65,73,83,88,96,98,107,110,113,121,122,131,136},{0,11,17,25,30,35,40,42,48,54,59,65,73,83,88,96,102,107,113,114,121,126,131,136},{0,11,17,25,30,40,42,48,54,59,65,71,73,83,88,96,102,113,114,121,126,131,136,143},{0,11,17,25,34,35,40,46,48,58,59,65,73,83,88,96,106,107,113,118,121,130,131,136},{0,11,17,25,34,40,46,48,58,59,65,71,73,83,88,96,106,113,118,121,130,131,136,143},{0,11,17,25,35,38,40,48,50,59,62,65,73,83,88,96,107,110,113,121,122,131,134,136},{0,11,17,25,35,40,42,48,54,59,65,66,73,83,88,96,107,113,114,121,126,131,136,138},{0,11,17,25,35,40,46,48,58,59,65,70,73,83,88,96,107,113,118,121,130,131,136,142},{0,11,18,23,26,33,35,40,45,48,57,66,74,83,88,95,96,105,107,114,117,122,129,136},{0,11,18,23,26,33,40,45,48,57,66,71,74,83,88,95,96,105,114,117,122,129,136,143},{0,11,18,23,26,35,40,45,48,57,66,69,74,83,88,95,96,107,114,117,122,129,136,141},{0,11,18,25,26,35,37,40,48,49,59,66,74,83,88,96,97,107,109,114,121,122,131,136},{0,11,18,26,29,35,40,41,48,53,59,66,74,83,88,96,101,107,113,114,122,125,131,136},{0,11,18,26,33,35,40,45,48,57,59,66,74,83,88,96,105,107,114,117,122,129,131,136},{0,11,18,26,33,40,45,48,57,59,66,71,74,83,88,96,105,114,117,122,129,131,136,143},{0,11,18,26,35,37,40,48,49,59,61,66,74,83,88,96,107,109,114,121,122,131,133,136},{0,11,18,26,35,40,41,48,53,59,65,66,74,83,88,96,107,113,114,122,125,131,136,137},{0,11,18,26,35,40,45,48,57,59,66,69,74,83,88,96,107,114,117,122,129,131,136,141},{0,11,19,22,25,30,37,40,48,49,59,67,70,78,88,96,97,107,109,115,118,121,126,136},{0,11,19,22,29,30,40,41,48,53,59,67,70,78,88,96,101,107,113,115,118,125,126,136},{0,11,19,22,30,33,40,45,48,57,59,67,70,78,88,96,105,107,115,117,118,126,129,136},{0,11,19,22,30,37,40,48,49,59,61,67,70,78,88,96,107,109,115,118,121,126,133,136},{0,11,19,22,30,40,41,48,53,59,65,67,70,78,88,96,107,113,115,118,125,126,136,137},{0,11,19,22,30,40,45,48,57,59,67,69,70,78,88,96,107,115,117,118,126,129,136,141},{0,11,19,24,25,30,37,38,48,49,59,67,72,78,86,96,97,107,109,115,120,121,126,134},{0,11,19,24,25,34,37,42,48,49,59,67,72,82,90,96,97,107,109,115,120,121,130,138},{0,11,19,24,26,33,38,41,48,50,59,67,72,81,89,96,98,107,110,115,120,122,129,137},{0,11,19,24,29,30,38,41,48,53,59,67,72,78,86,96,101,107,113,115,120,125,126,134},{0,11,19,24,29,34,41,42,48,53,59,67,72,82,90,96,101,107,113,115,120,125,130,138},{0,11,19,24,30,33,38,45,48,57,59,67,72,78,86,96,105,107,115,117,120,126,129,134},{0,11,19,24,30,33,41,42,48,54,59,67,72,81,89,96,102,107,114,115,120,126,129,137},{0,11,19,24,30,37,38,48,49,59,61,67,72,78,86,96,107,109,115,120,121,126,133,134},{0,11,19,24,30,38,41,48,53,59,65,67,72,78,86,96,107,113,115,120,125,126,134,137},{0,11,19,24,30,38,45,48,57,59,67,69,72,78,86,96,107,115,117,120,126,129,134,141},{0,11,19,24,33,34,41,46,48,58,59,67,72,81,89,96,106,107,115,118,120,129,130,137},{0,11,19,24,33,34,42,45,48,57,59,67,72,82,90,96,105,107,115,117,120,129,130,138},{0,11,19,24,33,38,41,48,50,59,62,67,72,81,89,96,107,110,115,120,122,129,134,137},{0,11,19,24,33,41,42,48,54,59,66,67,72,81,89,96,107,114,115,120,126,129,137,138},{0,11,19,24,33,41,46,48,58,59,67,70,72,81,89,96,107,115,118,120,129,130,137,142},{0,11,19,24,34,37,42,48,49,59,61,67,72,82,90,96,107,109,115,120,121,130,133,138},{0,11,19,24,34,41,42,48,53,59,65,67,72,82,90,96,107,113,115,120,125,130,137,138},{0,11,19,24,34,42,45,48,57,59,67,69,72,82,90,96,107,115,117,120,129,130,138,141},{0,11,19,25,26,33,38,40,48,50,59,67,73,81,88,96,98,107,110,115,121,122,129,136},{0,11,19,25,26,34,37,40,48,49,59,67,74,82,88,96,97,107,109,115,121,122,130,136},{0,11,19,25,30,33,40,42,48,54,59,67,73,81,88,96,102,107,114,115,121,126,129,136},{0,11,19,25,30,37,38,40,48,49,59,67,78,86,88,96,97,107,109,115,121,126,134,136},{0,11,19,25,33,34,40,46,48,58,59,67,73,81,88,96,106,107,115,118,121,129,130,136},{0,11,19,25,33,38,40,48,50,59,62,67,73,81,88,96,107,110,115,121,122,129,134,136},{0,11,19,25,33,40,42,48,54,59,66,67,73,81,88,96,107,114,115,121,126,129,136,138},{0,11,19,25,33,40,46,48,58,59,67,70,73,81,88,96,107,115,118,121,129,130,136,142},{0,11,19,25,34,37,42,48,49,59,60,67,72,82,90,97,107,109,115,120,121,130,132,138},{0,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18,130,131,133,136,139,141,142},{0,35,37,40,43,46,48,49,58,61,70,83,88,91,96,109,118,121,130,131,133,136,139,142},{0,35,37,40,45,46,47,48,58,59,70,85,88,93,96,107,118,119,130,131,133,136,141,142},{0,35,37,40,46,47,48,49,58,59,61,70,88,96,107,109,118,119,121,130,131,133,136,142},{0,35,37,42,43,45,48,54,60,66,72,83,85,91,93,114,120,126,131,132,133,138,139,141},{0,35,37,42,43,48,49,54,60,61,66,72,83,91,109,114,120,121,126,131,132,133,138,139},{0,35,37,42,45,47,48,54,59,60,66,72,85,93,107,114,119,120,126,131,132,133,138,141},{0,35,37,43,45,46,48,58,60,70,72,83,85,91,93,118,120,130,131,132,133,139,141,142},{0,35,37,43,46,48,49,58,60,61,70,72,83,91,109,118,120,121,130,131,132,133,139,142},{0,35,37,45,46,47,48,58,59,60,70,72,85,93,107,118,119,120,130,131,132,133,141,142},{0,35,38,40,41,43,46,48,53,65,83,86,88,91,94,96,113,125,131,134,136,137,139,142},{0,35,38,40,41,43,48,50,53,62,65,83,88,91,96,110,113,122,125,131,134,136,137,139},{0,35,38,40,41,46,47,48,53,59,65,86,88,94,96,107,113,119,125,131,134,136,137,142},{0,35,38,40,41,47,48,50,53,59,62,65,88,96,107,110,113,119,122,125,131,134,136,137},{0,35,38,40,43,45,46,48,57,69,83,86,88,91,94,96,117,129,131,134,136,139,141,142},{0,35,38,40,43,45,48,50,57,62,69,83,88,91,96,110,117,122,129,131,134,136,139,141},{0,35,38,40,45,46,47,48,57,59,69,86,88,94,96,107,117,119,129,131,134,136,141,142},{0,35,38,40,45,47,48,50,57,59,62,69,88,96,107,110,117,119,122,129,131,134,136,141},{0,35,38,41,43,46,48,53,60,65,72,83,86,91,94,113,120,125,131,132,134,137,139,142},{0,35,38,41,43,48,50,53,60,62,65,72,83,91,110,113,120,122,125,131,132,134,137,139},{0,35,38,41,46,47,48,53,59,60,65,72,86,94,107,113,119,120,125,131,132,134,137,142},{0,35,38,43,45,46,48,57,60,69,72,83,86,91,94,117,120,129,131,132,134,139,141,142},{0,35,38,43,45,48,50,57,60,62,69,72,83,91,110,117,120,122,129,131,132,134,139,141},{0,35,38,45,46,47,48,57,59,60,69,72,86,94,107,117,119,120,129,131,132,134,141,142},{0,35,40,41,42,43,48,53,54,65,66,83,88,91,96,113,114,125,126,131,136,137,138,139},{0,35,40,41,42,47,48,53,54,59,65,66,88,96,107,113,114,119,125,126,131,136,137,138},{0,35,40,41,43,46,48,53,58,65,70,83,88,91,96,113,118,125,130,131,136,137,139,142},{0,35,40,41,46,47,48,53,58,59,65,70,88,96,107,113,118,119,125,130,131,136,137,142},{0,35,40,42,43,45,48,54,57,66,69,83,88,91,96,114,117,126,129,131,136,138,139,141},{0,35,40,42,45,47,48,54,57,59,66,69,88,96,107,114,117,119,126,129,131,136,138,141},{0,35,40,43,45,46,48,57,58,69,70,83,88,91,96,117,118,129,130,131,136,139,141,142},{0,35,40,45,46,47,48,57,58,59,69,70,88,96,107,117,118,119,129,130,131,136,141,142},{0,35,41,42,43,48,53,54,60,65,66,72,83,91,113,114,120,125,126,131,132,137,138,139},{0,35,41,43,46,48,53,58,60,65,70,72,83,91,113,118,120,125,130,131,132,137,139,142},{0,35,42,43,45,48,54,57,60,66,69,72,83,91,114,117,120,126,129,131,132,138,139,141},{0,35,43,45,46,48,57,58,60,69,70,72,83,91,117,118,120,129,130,131,132,139,141,142},{0,37,38,39,40,45,46,48,51,63,85,86,88,93,94,96,111,123,133,134,135,136,141,142},{0,37,38,39,40,45,47,48,50,62,85,87,88,93,95,96,110,122,133,134,135,136,141,143},{0,37,38,39,40,45,48,50,51,62,63,85,88,93,96,110,111,122,123,133,134,135,136,141},{0,37,38,39,40,46,47,48,49,61,86,87,88,94,95,96,109,121,133,134,135,136,142,143},{0,37,38,39,40,46,48,49,51,61,63,86,88,94,96,109,111,121,123,133,134,135,136,142},{0,37,38,39,40,47,48,49,50,61,62,87,88,95,96,109,110,121,122,133,134,135,136,143},{0,37,38,39,40,48,49,50,51,61,62,63,88,96,109,110,111,121,122,123,133,134,135,136},{0,37,38,39,45,46,47,48,60,72,85,86,87,93,94,95,120,132,133,134,135,141,142,143},{0,37,38,39,45,46,48,51,60,63,72,85,86,93,94,111,120,123,132,133,134,135,141,142},{0,37,38,39,45,47,48,50,60,62,72,85,87,93,95,110,120,122,132,133,134,135,141,143},{0,37,38,39,45,48,50,51,60,62,63,72,85,93,110,111,120,122,123,132,133,134,135,141},{0,37,38,39,46,47,48,49,60,61,72,86,87,94,95,109,120,121,132,133,134,135,142,143},{0,37,38,39,46,48,49,51,60,61,63,72,86,94,109,111,120,121,123,132,133,134,135,142},{0,37,38,39,47,48,49,50,60,61,62,72,87,95,109,110,120,121,122,132,133,134,135,143},{0,37,38,40,43,45,46,48,55,67,85,86,88,93,94,96,115,127,133,134,136,139,141,142},{0,37,38,40,43,45,48,50,55,62,67,85,88,93,96,110,115,122,127,133,134,136,139,141},{0,37,38,40,43,46,48,49,55,61,67,86,88,94,96,109,115,121,127,133,134,136,139,142},{0,37,38,40,43,48,49,50,55,61,62,67,88,96,109,110,115,121,122,127,133,134,136,139},{0,37,38,40,45,46,47,48,59,71,85,86,88,93,94,96,119,131,133,134,136,141,142,143},{0,37,38,40,45,47,48,50,59,62,71,85,88,93,96,110,119,122,131,133,134,136,141,143},{0,37,38,40,46,47,48,49,59,61,71,86,88,94,96,109,119,121,131,133,134,136,142,143},{0,37,38,40,47,48,49,50,59,61,62,71,88,96,109,110,119,121,122,131,133,134,136,143},{0,37,38,43,45,46,48,55,60,67,72,85,86,93,94,115,120,127,132,133,134,139,141,142},{0,37,38,43,45,48,50,55,60,62,67,72,85,93,110,115,120,122,127,132,133,134,139,141},{0,37,38,43,46,48,49,55,60,61,67,72,86,94,109,115,120,121,127,132,133,134,139,142},{0,37,38,45,46,47,48,59,60,71,72,85,86,93,94,119,120,131,132,133,134,141,142,143},{0,37,38,45,47,48,50,59,60,62,71,72,85,93,110,119,120,122,131,132,133,134,141,143},{0,37,38,46,47,48,49,59,60,61,71,72,86,94,109,119,120,121,131,132,133,134,142,143},{0,37,39,40,42,45,47,48,54,66,85,87,88,93,95,96,114,126,133,135,136,138,141,143},{0,37,39,40,42,45,48,51,54,63,66,85,88,93,96,111,114,123,126,133,135,136,138,141},{0,37,39,40,42,47,48,49,54,61,66,87,88,95,96,109,114,121,126,133,135,136,138,143},{0,37,39,40,42,48,49,51,54,61,63,66,88,96,109,111,114,121,123,126,133,135,136,138},{0,37,39,40,45,46,47,48,58,70,85,87,88,93,95,96,118,130,133,135,136,141,142,143},{0,37,39,40,45,46,48,51,58,63,70,85,88,93,96,111,118,123,130,133,135,136,141,142},{0,37,39,40,46,47,48,49,58,61,70,87,88,95,96,109,118,121,130,133,135,136,142,143},{0,37,39,40,46,48,49,51,58,61,63,70,88,96,109,111,118,121,123,130,133,135,136,142},{0,37,39,42,45,47,48,54,60,66,72,85,87,93,95,114,120,126,132,133,135,138,141,143},{0,37,39,42,45,48,51,54,60,63,66,72,85,93,111,114,120,123,126,132,133,135,138,141},{0,37,39,42,47,48,49,54,60,61,66,72,87,95,109,114,120,121,126,132,133,135,138,143},{0,37,39,45,46,47,48,58,60,70,72,85,87,93,95,118,120,130,132,133,135,141,142,143},{0,37,39,45,46,48,51,58,60,63,70,72,85,93,111,118,120,123,130,132,133,135,141,142},{0,37,39,46,47,48,49,58,60,61,70,72,87,95,109,118,120,121,130,132,133,135,142,143},{0,37,40,42,43,45,48,54,55,66,67,85,88,93,96,114,115,126,127,133,136,138,139,141},{0,37,40,42,43,48,49,54,55,61,66,67,88,96,109,114,115,121,126,127,133,136,138,139},{0,37,40,42,45,47,48,54,59,66,71,85,88,93,96,114,119,126,131,133,136,138,141,143},{0,37,40,42,47,48,49,54,59,61,66,71,88,96,109,114,119,121,126,131,133,136,138,143},{0,37,40,43,45,46,48,55,58,67,70,85,88,93,96,115,118,127,130,133,136,139,141,142},{0,37,40,43,46,48,49,55,58,61,67,70,88,96,109,115,118,121,127,130,133,136,139,142},{0,37,40,45,46,47,48,58,59,70,71,85,88,93,96,118,119,130,131,133,136,141,142,143},{0,37,40,46,47,48,49,58,59,61,70,71,88,96,109,118,119,121,130,131,133,136,142,143},{0,37,42,43,45,48,54,55,60,66,67,72,85,93,114,115,120,126,127,132,133,138,139,141},{0,37,42,45,47,48,54,59,60,66,71,72,85,93,114,119,120,126,131,132,133,138,141,143},{0,37,43,45,46,48,55,58,60,67,70,72,85,93,115,118,120,127,130,132,133,139,141,142},{0,37,45,46,47,48,58,59,60,70,71,72,85,93,118,119,120,130,131,132,133,141,142,143},{0,38,39,40,41,46,47,48,53,65,86,87,88,94,95,96,113,125,134,135,136,137,142,143},{0,38,39,40,41,46,48,51,53,63,65,86,88,94,96,111,113,123,125,134,135,136,137,142},{0,38,39,40,41,47,48,50,53,62,65,87,88,95,96,110,113,122,125,134,135,136,137,143},{0,38,39,40,41,48,50,51,53,62,63,65,88,96,110,111,113,122,123,125,134,135,136,137},{0,38,39,40,45,46,47,48,57,69,86,87,88,94,95,96,117,129,134,135,136,141,142,143},{0,38,39,40,45,46,48,51,57,63,69,86,88,94,96,111,117,123,129,134,135,136,141,142},{0,38,39,40,45,47,48,50,57,62,69,87,88,95,96,110,117,122,129,134,135,136,141,143},{0,38,39,40,45,48,50,51,57,62,63,69,88,96,110,111,117,122,123,129,134,135,136,141},{0,38,39,41,46,47,48,53,60,65,72,86,87,94,95,113,120,125,132,134,135,137,142,143},{0,38,39,41,46,48,51,53,60,63,65,72,86,94,111,113,120,123,125,132,134,135,137,142},{0,38,39,41,47,48,50,53,60,62,65,72,87,95,110,113,120,122,125,132,134,135,137,143},{0,38,39,45,46,47,48,57,60,69,72,86,87,94,95,117,120,129,132,134,135,141,142,143},{0,38,39,45,46,48,51,57,60,63,69,72,86,94,111,117,120,123,129,132,134,135,141,142},{0,38,39,45,47,48,50,57,60,62,69,72,87,95,110,117,120,122,129,132,134,135,141,143},{0,38,40,41,43,46,48,53,55,65,67,86,88,94,96,113,115,125,127,134,136,137,139,142},{0,38,40,41,43,48,50,53,55,62,65,67,88,96,110,113,115,122,125,127,134,136,137,139},{0,38,40,41,46,47,48,53,59,65,71,86,88,94,96,113,119,125,131,134,136,137,142,143},{0,38,40,41,47,48,50,53,59,62,65,71,88,96,110,113,119,122,125,131,134,136,137,143},{0,38,40,43,45,46,48,55,57,67,69,86,88,94,96,115,117,127,129,134,136,139,141,142},{0,38,40,43,45,48,50,55,57,62,67,69,88,96,110,115,117,122,127,129,134,136,139,141},{0,38,40,45,46,47,48,57,59,69,71,86,88,94,96,117,119,129,131,134,136,141,142,143},{0,38,40,45,47,48,50,57,59,62,69,71,88,96,110,117,119,122,129,131,134,136,141,143},{0,38,41,43,46,48,53,55,60,65,67,72,86,94,113,115,120,125,127,132,134,137,139,142},{0,38,41,46,47,48,53,59,60,65,71,72,86,94,113,119,120,125,131,132,134,137,142,143},{0,38,43,45,46,48,55,57,60,67,69,72,86,94,115,117,120,127,129,132,134,139,141,142},{0,38,45,46,47,48,57,59,60,69,71,72,86,94,117,119,120,129,131,132,134,141,142,143},{0,39,40,41,42,47,48,53,54,65,66,87,88,95,96,113,114,125,126,135,136,137,138,143},{0,39,40,41,42,48,51,53,54,63,65,66,88,96,111,113,114,123,125,126,135,136,137,138},{0,39,40,41,46,47,48,53,58,65,70,87,88,95,96,113,118,125,130,135,136,137,142,143},{0,39,40,41,46,48,51,53,58,63,65,70,88,96,111,113,118,123,125,130,135,136,137,142},{0,39,40,42,45,47,48,54,57,66,69,87,88,95,96,114,117,126,129,135,136,138,141,143},{0,39,40,42,45,48,51,54,57,63,66,69,88,96,111,114,117,123,126,129,135,136,138,141},{0,39,40,45,46,47,48,57,58,69,70,87,88,95,96,117,118,129,130,135,136,141,142,143},{0,39,40,45,46,48,51,57,58,63,69,70,88,96,111,117,118,123,129,130,135,136,141,142},{0,39,41,42,47,48,53,54,60,65,66,72,87,95,113,114,120,125,126,132,135,137,138,143},{0,39,41,46,47,48,53,58,60,65,70,72,87,95,113,118,120,125,130,132,135,137,142,143},{0,39,42,45,47,48,54,57,60,66,69,72,87,95,114,117,120,126,129,132,135,138,141,143},{0,39,45,46,47,48,57,58,60,69,70,72,87,95,117,118,120,129,130,132,135,141,142,143},{0,40,41,42,43,48,53,54,55,65,66,67,88,96,113,114,115,125,126,127,136,137,138,139},{0,40,41,42,47,48,53,54,59,65,66,71,88,96,113,114,119,125,126,131,136,137,138,143},{0,40,41,43,46,48,53,55,58,65,67,70,88,96,113,115,118,125,127,130,136,137,139,142},{0,40,41,46,47,48,53,58,59,65,70,71,88,96,113,118,119,125,130,131,136,137,142,143},{0,40,42,43,45,48,54,55,57,66,67,69,88,96,114,115,117,126,127,129,136,138,139,141},{0,40,42,45,47,48,54,57,59,66,69,71,88,96,114,117,119,126,129,131,136,138,141,143},{0,40,43,45,46,48,55,57,58,67,69,70,88,96,115,117,118,127,129,130,136,139,141,142},{0,40,45,46,47,48,57,58,59,69,70,71,88,96,117,118,119,129,130,131,136,141,142,143}};

NN={0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143};


H=Array[{0}&,Length[T1[[1]]]+1   ];
Exc=Array[{0}&,10000];

K=T1[[1]];
dd=1;
For[t1=1, t1≤Length[T1], K=T1[[t1]];
For[t2=1,t2≤Length[T2],B=T2[[t2]];H[[1]]={0};
For[j=1, j≤Length[K], 
H[[j+1]]=Mod[(K[[j]]+B)⋃ H[[j]],n];
j++];
If[Not[H[[Length[K]+1]]⋃{}  == NN],Exc[[dd]]={T1[[t1]],T2[[t2]]}; dd=dd+1];
t2++];
t1++]; 

Exc=Drop[Exc⋃{},1];

Length[Exc]

Clear[H];

(*Here comes the Fourier selection for T2*)

Dynamic[{j,v,vvv,vvvv}]

 
Length[T2]

n=144;
H=Array[0&,{Length[T2],n}];
MF=Array[0&,100];
MT=Array[0&,100];

For[j=1, j≤Length[T2], 
For[k=1, k≤n,
If[MemberQ[T2[[j]],Mod[k,n]],H[[j,k]]=1];
k++];
j++];

MF[[1]]=H[[1]];
MT[[1]]=T2[[1]];

cont=1;
v=1;
For[j=2, j≤Length[T2], cont=1;
For[c=1,c≤v,
For[k=1, k≤n,
If[(Not[Chop[Fourier[MF[[c]]]][[k]]==Chop[Fourier[H[[j]]]][[k]]])∧(Chop[Fourier[MF[[c]]]][[k]]==0∨Chop[Fourier[H[[j]]]][[k]]==0), 
k=n+1;cont=cont+1];
k++];
If[cont≤c, c=v+1];
c++];
If[cont==v+1, MF[[v+1]]=H[[j]];MT[[v+1]]=T2[[j]]; v=v+1];
j++]; 


MT=Drop[MT⋃{},1];
MF=Drop[MF⋃{},1];
Length[MF]



MT

(*{{3,8,16},{2,4,9}};*)

(* Next, we see what those Fourier-zero sets are. *)

n=144;
J=MF;
F1=Array[1&,{Length[J],n}];

For[i=1,i<=Length[J],
For[j=1,j≤n,
If[Chop[Fourier[J[[i]]]][[j]]==0, F1[[i,j]]=0];
j++];
i++];

F1;


n=144;
FF1=Array[0&,{Length[F1],n}];

For[i=1,i≤Length[F1],
For[j=1,j≤n,
If[F1[[i,j]]==0, FF1[[i,j]]=j];
j++];
i++];

For[j=1,j≤Length[FF1],
FF1[[j]]=Drop[FF1[[j]]⋃{},1];
j++];
FF1;


FF=FF1-1;

For[m=1,m≤Length[FF1],
For[j=1,j≤Length[FF1[[m]]],
FF[[m,j]]=GCD[FF[[m,j]],n];
j++];
m++];
FF;
For[j=1,j≤Length[FF],
FF[[j]]=FF[[j]]⋃{};
j++];
FF

FT=n/FF
CFT=Array[0&,Length[FT]];
For[j=1,j≤Length[FT],
CFT[[j]]=Complement[Divisors[n],FT[[j]]];
j++];
CFT





(*And now the Fourier selection for T1*)


Dynamic[{j,v,vvv,vvvv}]

T2=T1;

Clear[H];

Length[T2]

n=144;
H=Array[0&,{Length[T2],n}];
MF=Array[0&,100];
MT=Array[0&,100];

For[j=1, j≤Length[T2], 
For[k=1, k≤n,
If[MemberQ[T2[[j]],Mod[k,n]],H[[j,k]]=1];
k++];
j++];

MF[[1]]=H[[1]];
MT[[1]]=T2[[1]];

cont=1;
v=1;
For[j=2, j≤Length[T2], cont=1;
For[c=1,c≤v,
For[k=1, k≤n,
If[(Not[Chop[Fourier[MF[[c]]]][[k]]==Chop[Fourier[H[[j]]]][[k]]])∧(Chop[Fourier[MF[[c]]]][[k]]==0∨Chop[Fourier[H[[j]]]][[k]]==0), 
k=n+1;cont=cont+1];
k++];
If[cont≤c, c=v+1];
c++];
If[cont==v+1, MF[[v+1]]=H[[j]];MT[[v+1]]=T2[[j]]; v=v+1];
j++]; 


MT=Drop[MT⋃{},1];
MF=Drop[MF⋃{},1];
Length[MF]



MT

(*{{3,8,16},{2,4,9}};*)

(* Next, we see what those Fourier-zero sets are. *)

n=144;
J=MF;
F1=Array[1&,{Length[J],n}];

For[i=1,i<=Length[J],
For[j=1,j≤n,
If[Chop[Fourier[J[[i]]]][[j]]==0, F1[[i,j]]=0];
j++];
i++];

F1;


n=144;
FF1=Array[0&,{Length[F1],n}];

For[i=1,i≤Length[F1],
For[j=1,j≤n,
If[F1[[i,j]]==0, FF1[[i,j]]=j];
j++];
i++];

For[j=1,j≤Length[FF1],
FF1[[j]]=Drop[FF1[[j]]⋃{},1];
j++];
FF1;


FF=FF1-1;

For[m=1,m≤Length[FF1],
For[j=1,j≤Length[FF1[[m]]],
FF[[m,j]]=GCD[FF[[m,j]],n];
j++];
m++];
FF;
For[j=1,j≤Length[FF],
FF[[j]]=FF[[j]]⋃{};
j++];
FF

FT=n/FF
CFT=Array[0&,Length[FT]];
For[j=1,j≤Length[FT],
CFT[[j]]=Complement[Divisors[n],FT[[j]]];
j++];
CFT

 
(*Finally we run the fill-out procedure for different representatives of T2*)

(*{{2,4,9,16},{3,8}};*)

Dynamic[{j,i,g, mark}]
Clear[n,B,R,Tile,A,U,mark,T,g,x];

n=144;

B={0,9,17,26,31,34,40,43,48,55,57,65,74,82,88,96,103,105,113,115,122,127,130,136}
Length[B];

Tile=Array[0&,1000000];
R=Array[0&,n];
A=Array[200&,{n/Length[B],n,Length[B]}];
U=Array[0&,n/Length[B]];
U[[1]]=B;
mark=Array[1&,n/Length[B]+1];
T=Array[{0}&,n/Length[B]];
g=1;

For[i=1, i>0,
If[Length[U[[i]]]==n,  Tile[[g]]=T[[i]]⋃{};g=g+1;T[[i]]=T[[i-1]];mark[[i]]=1;A[[i]]=Array[200&,{n,Length[B]}];i=i-1,
For[j=1, j≤n,k=100;
If[Not[MemberQ[U[[i]],Mod[j,n]]], k=0;
For[m=1, m≤Length[B],
If[U[[i]]⋂ Mod[(j-B[[m]])+B,n]=={}, k=k+1; A[[i,j,m]]=Mod[j-B[[m]],n]];
m++]];
R[[j]]=k;
j++];
min=Min[R]; 
If[(Not[min==0]),
t=Catch[Do[If[R[[ii]]==min, Throw[ii]],{ii,n}]];
If[mark[[i]]≤Length[B],
For[b=mark[[i]], b<=Length[B],mark[[i]]=b+1;z=0;
If[Not[A[[i,t,b]]==200], U[[i+1]]=U[[i]]⋃Mod[A[[i,t,b]]+B,n]; T[[i+1]]=T[[i]]⋃{A[[i,t,b]]}; b=Length[B]+1;z=1;i=i+1;];
b++];
If[z==0, mark[[i]]=1;A[[i]]=Array[200&,{n,Length[B]}];i=i-1];
, mark[[i]]=1;A[[i]]=Array[200&,{n,Length[B]}];T[[i]]=T[[i-1]];i=i-1];
, mark[[i]]=1;A[[i]]=Array[200&,{n,Length[B]}];T[[i]]=T[[i-1]];i=i-1];
];
];

Tile=Drop[Tile⋃{},1];
Length[Tile]
Tile;

H=Array[{0}&,Length[Tile[[1]]]+1];
K=Tile[[3]];
For[j=1, j≤Length[K], 
H[[j+1]]=Mod[(K[[j]]+B)⋃ H[[j]],n];
j++];
H[[Length[K]+1]]⋃{} 

For[j=1, j≤Length[Tile], P=Tile[[j]];
For[i=2, i<=Length[P], 
If[MemberQ[Tile,Mod[P-P[[i]],n]⋃{}], Tile[[j]]=0;i=Length[P]+1];
i++];
j++]; 
Tile=Drop[Tile⋃{},1];
Length[Tile]
Tile







(*{{2,4,9,16},{3,8}};*)

Dynamic[{j,i,g, mark}]
Clear[n,B,R,Tile,A,U,mark,T,g,x];

n=144;

(*B=Array[0&,10];

For[i=1, i≤Length[B],
B[[i]]=RandomInteger[n-1];
i++];
B⋃{}*)

B={0,11,22,29,30,35,40,41,48,53,59,70,78,83,88,96,101,107,113,118,125,126,131,136}
Length[B];

Tile=Array[0&,1000000];
R=Array[0&,n];
A=Array[200&,{n/Length[B],n,Length[B]}];
U=Array[0&,n/Length[B]];
U[[1]]=B;
mark=Array[1&,n/Length[B]+1];
T=Array[{0}&,n/Length[B]];
g=1;

For[i=1, i>0,
If[Length[U[[i]]]==n,  Tile[[g]]=T[[i]]⋃{};g=g+1;T[[i]]=T[[i-1]];mark[[i]]=1;A[[i]]=Array[200&,{n,Length[B]}];i=i-1,
For[j=1, j≤n,k=100;
If[Not[MemberQ[U[[i]],Mod[j,n]]], k=0;
For[m=1, m≤Length[B],
If[U[[i]]⋂ Mod[(j-B[[m]])+B,n]=={}, k=k+1; A[[i,j,m]]=Mod[j-B[[m]],n]];
m++]];
R[[j]]=k;
j++];
min=Min[R]; 
If[(Not[min==0]),
t=Catch[Do[If[R[[ii]]==min, Throw[ii]],{ii,n}]];
If[mark[[i]]≤Length[B],
For[b=mark[[i]], b<=Length[B],mark[[i]]=b+1;z=0;
If[Not[A[[i,t,b]]==200], U[[i+1]]=U[[i]]⋃Mod[A[[i,t,b]]+B,n]; T[[i+1]]=T[[i]]⋃{A[[i,t,b]]}; b=Length[B]+1;z=1;i=i+1;];
b++];
If[z==0, mark[[i]]=1;A[[i]]=Array[200&,{n,Length[B]}];i=i-1];
, mark[[i]]=1;A[[i]]=Array[200&,{n,Length[B]}];T[[i]]=T[[i-1]];i=i-1];
, mark[[i]]=1;A[[i]]=Array[200&,{n,Length[B]}];T[[i]]=T[[i-1]];i=i-1];
];
];

Tile=Drop[Tile⋃{},1];
Length[Tile]
Tile;

H=Array[{0}&,Length[Tile[[1]]]+1];
K=Tile[[3]];
For[j=1, j≤Length[K], 
H[[j+1]]=Mod[(K[[j]]+B)⋃ H[[j]],n];
j++];
H[[Length[K]+1]]⋃{} 

For[j=1, j≤Length[Tile], P=Tile[[j]];
For[i=2, i<=Length[P], 
If[MemberQ[Tile,Mod[P-P[[i]],n]⋃{}], Tile[[j]]=0;i=Length[P]+1];
i++];
j++]; 
Tile=Drop[Tile⋃{},1];
Length[Tile]
Tile