The following code is a slight modification of the code to find the 16 possible betti tables. Simply go thru all the tables, but only resolve the quadratic terms.
i1 : GetQuads = (d,k,n)->(
R=ZZ/2[x_1..x_k];
quartics=super basis(d,R);
MonList=apply(rank source quartics, i->quartics_(0,i));
L=subsets(MonList,n);
J=apply(L, j->(F=gens ideal sum j;
InvSysF=fromDual F;
Idegs=degrees source mingens ideal InvSysF;
if (not ((member({1},Idegs)))) then minimalBetti coker super basis(2, ideal InvSysF)));
Jlist =drop(unique J,1);
netList pack(4,Jlist))
o1 = GetQuads
o1 : FunctionClosure
|
i2 : GetQuads(4,4,2)
+--------------+--------------+--------------+----------------+
| 0 1 2 3| 0 1 2 3| 0 1 2 3| 0 1 2 3 |
o2 = |total: 1 6 8 3|total: 1 5 5 1|total: 1 4 4 1|total: 1 4 4 1 |
| 0: 1 . . .| 0: 1 . . .| 0: 1 . . .| 0: 1 . . . |
| 1: . 6 8 3| 1: . 5 5 .| 1: . 4 3 .| 1: . 4 4 1 |
| | 2: . . . 1| 2: . . 1 1| |
+--------------+--------------+--------------+----------------+
| 0 1 2 3| 0 1 2 | 0 1 2 3| 0 1 2 3 4|
|total: 1 4 5 2|total: 1 3 2 |total: 1 3 3 1|total: 1 4 6 4 1|
| 0: 1 . . .| 0: 1 . . | 0: 1 . . .| 0: 1 . . . .|
| 1: . 4 2 .| 1: . 3 2 | 1: . 3 . .| 1: . 4 . . .|
| 2: . . 3 2| | 2: . . 3 .| 2: . . 6 . .|
| | | 3: . . . 1| 3: . . . 4 .|
| | | | 4: . . . . 1|
+--------------+--------------+--------------+----------------+
| 0 1 2 | 0 1 2 3| 0 1 2 3| |
|total: 1 2 1 |total: 1 5 6 2|total: 1 5 6 2| |
| 0: 1 . . | 0: 1 . . .| 0: 1 . . .| |
| 1: . 2 . | 1: . 5 5 1| 1: . 5 6 2| |
| 2: . . 1 | 2: . . 1 1| | |
+--------------+--------------+--------------+----------------+
|