Three-dimensional symmetric designs of propriety 3

A three-dimensional (v,k,λ) designs of propriety 3 is a v×v×v array with {0,1}-entries such that every v×v subarray (layer) contains exactly k entries 1, and the scalar product of any pair of parallel layers is λ. These objects are studied in the paper [1]; similar objects were previously studied in [2, 3, 4]. This web page contains online versions of the tables from [1], with links to files containing the actual cubes and difference sets in GAP-readable format. The GAP package PAG contains functions for working with the objects. All files in a single archive can be downloaded here.

The first table contains the results of a complete enumeration of small C₃³(v,k,λ)-cubes up to equivalence. Each link points to a GAP-file with a list of cubes assigned to the variable c. For example, the C₃³(4,4,0)-cubes are read into GAP and tested by typing the following commands.

gap> LoadPackage("PAG");
gap> Read("4-4.cube");
gap> List(c,Cube3Test);
[ [ 4, [ 4 ], [ 0 ] ], [ 4, [ 4 ], [ 0 ] ] ]


v	k	λ	No. cubes

3	2	0	2
3	3	0	1
		1	1
3	4	1	1

4	2	0	8
4	3	0	4
4	4	0	2
4	5	λ	0
4	6	2	117
4	7	2	4
		3	2
4	8	4	19

5	2	0	23
5	3	0	251
5	4	0	40

6	2	0	157

Table 1. Numbers of inequivalent C₃³(v,k,λ)-cubes.

Tables 2 and 3 contain small cubes constructed from difference sets of propriety 3. The underlying groups are from the GAP Small Groups library. Here is an example of how to read (7,13,2) difference sets, transform them to cubes and test them using PAG-commands.

gap> Read("C7-13-2.ds");
gap> g:=SmallGroup(7,1);
<pc group of size 7 with 1 generator>
gap> c:=List(ds,x->OrthogonalArrayToCube(Development3(g,x)));;
gap> Collected(List(c,Cube3Test));
[ [ [ 7, [ 13 ], [ 2 ] ], 5 ] ]


v	k	λ	C_v

3	2	0	1
3	3	0	1
		1	1
3	4	1	1

5	2	0	1
5	3	0	2
5	4	0	1
5	5	0	1
		1	1
5	6	1	4
5	7	1	1
		2	6
5	8	2	2
5	9	2	2
		3	2
		4	1
5	10	3	6
		4	15
5	11	4	14
		5	7
5	12	5	5
		6	1

7	2	0	2
7	3	0	7
7	4	0	10
7	5	0	9
7	6	0	4
		1	5
7	7	0	2
		1	26
7	8	1	104
		2	1
7	9	1	158
		2	11
7	10	1	24
		2	152
		3	1
7	11	2	1258
		3	11
7	12	2	659
		3	542
		4	1
7	13	2	5
		3	3875
		4	56
7	14	3	1943
		4	2021
		5	17

Table 2. Numbers of inequivalent C₃³(v,k,λ)-cubes coming from cyclic difference sets.


v	k	λ	C₄	C₂×C₂	Total

4	2	0	1	1	2
4	3	0	1	1	2
4	4	0	0	1	1
4	5	λ	0	0	0
4	6	2	5	6	7
4	7	2	1	1	1
		3	0	0	0
4	8	4	0	1	1


v	k	λ	C₆	S₃	Total

6	2	0	3	2	4
6	3	0	7	4	9
6	4	0	5	4	9
6	5	0	2	1	3
6	6	λ	0	0	0
6	7	λ	0	0	0
6	8	2	10	3	13
6	9	2	158	42	200
6	10	2	71	12	83
6	11	2	2	0	2
		4	3	0	3
6	12	4	283	42	322
6	13	4	91	13	99
6	14	4	9	1	9
		6	58	10	66
6	15	6	147	29	160
6	16	6	109	7	116
		8	10	6	14
6	17	8	180	17	193
6	18	8	264	30	292
		10	4	2	4

Table 3. Numbers of inequivalent C₃³(v,k,λ)-cubes coming from difference sets.

The fourth table contains (7,k,λ) difference sets of propriety 3 with multipliers of order 2 and 3. For 15 ≤ k ≤ 24, a complete enumeration of difference sets was out of reach. The computation is much faster if nontrivial multipliers are assumed.


v	k	λ	C₇⋊C₂	C₇⋊C₃	Total

7	15	3	0	5	5
		4	0	63	63
		6	0	2	2
7	16	4	0	58	58
		5	0	4	4
		6	1	1	2
7	17	6	1	0	1
7	18	5	0	24	24
		6	3	5	5
		7	1	36	36
7	19	6	3	82	82
		7	0	3	3
		8	0	2	2
		9	1	1	1
7	20	λ	0	0	0
7	21	7	0	10	10
		8	0	17	17
		9	0	57	57
		11	0	2	2
7	22	8	0	11	11
		9	0	90	90
		11	0	9	9
7	23	λ	0	0	0
7	24	10	0	45	45
		11	2	3	3
		12	2	55	55
		13	0	1	1

Table 4. Numbers of inequivalent C₃³(7,k,λ)-cubes coming from difference sets with multipliers.

References

A. Bahmanian, V. Krcadinac, L. Relic, S. Suda, Three-dimensional symmetric designs of propriety 3, preprint, 2025. http://arxiv.org/abs/2510.17337
V. Krcadinac, M. O. Pavcevic, K. Tabak, Cubes of symmetric designs, Ars Math. Contemp. 25 (2025), no. 1, Paper No. 10, 16 pp. https://doi.org/10.26493/1855-3974.3222.e53
V. Krcadinac, L. Relic, Projection cubes of symmetric designs, to appear in Math. Comput. Sci. (2025). https://arxiv.org/abs/2411.06936
V. Krcadinac, M. O. Pavcevic, On higher-dimensional symmetric designs, to appear in Exp. Math. (2025). https://doi.org/10.1080/10586458.2025.2521450

Vedran Krcadinac,