Projection cubes of symmetric designs

Projection cubes of symmetric designs are introduced in the paper [5]. A similar concept was earlier studied in [4]. A (v,k,λ) projection n-cube is an n-dimensional matrix with {0,1}-entries such that every 2-dimensional projection is an incidence matrix of a symmetric (v,k,λ) design. The set of all such objects is denoted Pn(v,k,λ). Here are pictures of three P3(7,3,1)-cubes rendered in POV-Ray:

C1
C2
C3

These are the cubes C1, C2, and C3 from [5]. The file examples.oa contains all examples from the paper in GAP-compatible format. They can be explored using commands from the PAG package. Instructions and examples are available in the PAG manual. The following table contains lower bounds on the number of inequivalent cubes in Pn(v,k,λ) with link to files in GAP format. These examples were constructed using n-dimensional difference sets, see [5].

(v,k,λ)n
 3  4  5  6   7  8  9  10   11  12  13  14 
(7,3,1)22 111
(7,4,2)22 111
(11,5,2)24 664 211 1
(11,6,3)24 664 211 1
(13,4,1)37 101414 1073 111
(15,7,3)3
(15,8,4)61
  (16,6,2)   724   8464  1601   1754  986   505  178   70  16   7  2   1 
(21,5,1)6

There are at least 102 inequivalent projection cubes in P3(16,6,2) that cannot be obtained from difference sets. They were constructed by prescribing autotopy groups and using a Kramer-Mesner-like procedure [5]. A GAP file with these examples is available here.

References

  1. The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.13.1, 2024. https://www.gap-system.org
  2. V. Krcadinac, Prescribed Automorphism Groups, Version 0.2.4, 2024 (GAP package). https://vkrcadinac.github.io/PAG
  3. V. Krcadinac, The PAG manual, 2024. https://web.math.pmf.unizg.hr/acco/PAGmanual.pdf
  4. V. Krcadinac, M. O. Pavcevic, K. Tabak, Cubes of symmetric designs, Ars Math. Contemp. (2024). https://doi.org/10.26493/1855-3974.3222.e53
  5. V. Krcadinac, L. Relic, Projection cubes of symmetric designs, preprint, 2024. https://arxiv.org/abs/2411.06936
  6. Persistence of Vision Raytracer, Version 3.7, 2013. http://www.povray.org

Vedran Krcadinac,