Cubes of symmetric designs are a special case of proper n-dimensional combinatorial designs from [1]. They are studied in the paper [3]. A complete classification of 3-dimensional group cubes of (16,6,2) designs was performed: there are 27 difference cubes and 946 non-difference group cubes up to equivalence. In addition, there are at least 1423 non-group cubes of (16,6,2) designs. Here is a table with links to gnu-zipped files containing the cubes in GAP format.
| Group (16,6,2) cubes | Non-group (16,6,2) cubes | |
| Difference | Non-difference | |
| 27 | 946 | ≥ 1423 |
The next table contains numbers of inequivalent group cubes of small orders. The column Nds contains numbers of inequivalent difference sets according to [4]. The numbers of difference cubes are given in the column Ndc, with links to files of difference sets giving rise to inequivalent 3-cubes. See page 16 of the PAG package manual [2] for how these numbers were computed.
The column Ngc contains lower bounds for numbers of non-difference group cubes,
or the exact number for (21,5,1). The numbers are linked to files containing
symmetric designs with difference sets as blocks, which are not developments
of their blocks. Let dev be the list of difference sets linked in
the Ndc column, and nondev the list of designs linked in the Ngc
column. Then the corresponding difference and non-difference group 3-cubes are
obtained by the following GAP commands, where v is the number of
points:
dc:=List(dev,x->DifferenceCube(SmallGroup(v,x[1]),x[2],3));
gc:=List(nondev,x->GroupCube(SmallGroup(v,x[1]),x[2],3));
See pages 17–19 of the PAG manual [2] for a more detailed explanation.
| Parameters | Nds | Ndc | Ngc |
| (21,5,1) | 2 | 2 | 1 |
| (27,13,6) | 3 | 2 | ≥ 7 |
| (36,15,6) | 35 | 35 | ≥ 373 |
| (45,12,3) | 2 | 2 | ≥ 6 |
| (63,31,15) | 6 | 6 | ≥ 9 |
| (64,28,12) | 330159 | < 330159 | ≥ 1 |
| (96,20,4) | 2627 | 1806 | ≥ 1 |
Vedran Krcadinac,