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PBIB-designs and association schemes arising from minimum bi-connected dominating sets of hypercubes

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Abstract

In this paper we disprove a result on the PBIB designs arising from \(\gamma _{bc}\)-sets in \(Q_{4}\), with \(m = 4\) association schemes proved by Chaluvaraju et al. (Afr Math 29:47–63, 2018). In Ref. [11], the authors had posed an open problem to find the PBIB designs in hypercubes of dimension higher than 5. Here, we settle the problem for the first few values, and give a general iterative proof technique for higher dimensional hypercubes.

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Acknowledgements

We thank the unknown referees for the suggestions that has improved the overall presentation of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Bangalore University, Central College Campus, Bengaluru, 560 001, India

    Medha Itagi Huilgol & M. D. Vidya

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  1. Medha Itagi Huilgol
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  2. M. D. Vidya
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Correspondence to Medha Itagi Huilgol.

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Appendix A: List of all 1152 \(\gamma _{bc}\)-sets of \(Q_{5}\)

Appendix A: List of all 1152 \(\gamma _{bc}\)-sets of \(Q_{5}\)

$$\begin{aligned} \begin{array}{llll} &{}\lbrace 32,29,9,12,16,4,3,2,22,23\rbrace &{}\lbrace 32,31,11,12,16,4,1,2,22,21\rbrace &{}\lbrace 32,31,11,12,16,13,1,2,22,21\rbrace \\ &{}\lbrace 32,17,8,12,16,13,14,2,22,27\rbrace &{}\lbrace 32,17,8,12,16,15,14,2,22,27\rbrace &{}\lbrace 32,29,9,12,16,15,3,2,22,23\rbrace \\ &{}\lbrace 32,29,9,12,8,4,3,2,22,23\rbrace &{}\lbrace 32,31,11,12,8,4,1,2,22,21\rbrace &{}\lbrace 32,25,16,12,8,7,6,2,22,19\rbrace \\ &{}\lbrace 32,29,9,12,8,7,3,2,22,23\rbrace &{}\lbrace 32,31,11,12,8,5,1,2,22,21\rbrace &{}\lbrace 32,25,16,12,8,5,6,2,22,19\rbrace \\ &{}\lbrace 32,17,8,12,11,10,14,2,22,27\rbrace &{}\lbrace 32,25,16,12,11,10,6,2,22,19\rbrace &{}\lbrace 32,29,9,12,11,7,3,2,22,23\rbrace \\ &{}\lbrace 32,17,8,12,9,10,14,2,22,27\rbrace &{}\lbrace 32,25,16,12,11,7,6,2,22,19\rbrace &{}\lbrace 32,29,9,12,11,15,3,2,22,23\rbrace \\ &{}\lbrace 32,17,8,12,11,15,14,2,22,27\rbrace &{}\lbrace 32,31,11,12,9,5,1,2,22,21\rbrace &{}\lbrace 32,25,16,12,9,5,6,2,22,19\rbrace \\ &{}\lbrace 32,31,11,12,9,13,1,2,22,21\rbrace &{}\lbrace 32,17,8,12,9,13,14,2,22,27\rbrace &{}\lbrace 32,25,16,12,9,10,6,2,22,19\rbrace \\ &{}\lbrace 25,28,13,16,4,3,7,6,19,18\rbrace &{}\lbrace 25,32,12,16,4,3,2,6,19,22\rbrace &{}\lbrace 25,26,15,16,4,1,5,6,19,20\rbrace \\ &{}\lbrace 25,32,12,16,4,1,2,6,19,22\rbrace &{}\lbrace 25,28,13,16,4,8,7,6,19,18\rbrace &{}\lbrace 25,26,15,16,4,8,5,6,19,20\rbrace \\ &{}\lbrace 25,26,15,16,13,1,5,6,19,20\rbrace &{}\lbrace 25,32,12,16,13,1,2,6,19,22\rbrace &{}\lbrace 25,26,15,16,13,9,5,6,19,20\rbrace \\ &{}\lbrace 25,24,4,16,13,9,10,6,19,30\rbrace &{}\lbrace 25,32,12,16,13,14,2,6,19,22\rbrace &{}\lbrace 25,24,4,16,13,14,10,6,19,30\rbrace \\ &{}\lbrace 25,32,12,16,15,14,2,6,19,22\rbrace &{}\lbrace 25,24,4,16,15,14,10,6,19,30\rbrace &{}\lbrace 25,28,13,16,15,11,7,6,19,18\rbrace \\ &{}\lbrace 25,24,4,16,15,11,10,6,19,30\rbrace &{}\lbrace 25,32,12,16,15,3,2,6,19,22\rbrace &{}\lbrace 25,28,13,16,15,3,7,6,19,18 \rbrace \\ &{}\lbrace 25,28,13,16,12,8,7,6,19,18\rbrace &{}\lbrace 25,26,15,16,12,8,5,6,19,20\rbrace &{}\lbrace 25,26,15,16,12,9,5,6,19,20\rbrace \\ &{}\lbrace 25,24,4,16,12,9,10,6,19,30\rbrace &{}\lbrace 25,28,13,16,12,11,7,6,19,18\rbrace &{}\lbrace 25,24,4,16,12,11,10,6,19,30\rbrace \\ &{}\lbrace 24,25,16,4,3,2,6,10,30,19\rbrace &{}\lbrace 24,17,8,4,3,2,14,10,30,27\rbrace &{}\lbrace 24,25,16,4,3,7,6,10,30,19\rbrace \\ &{}\lbrace 24,21,1,4,3,7,11,10,30,31\rbrace &{}\lbrace 24,21,1,4,3,15,11,10,30,31\rbrace &{}\lbrace 24,17,8,4,3,15,14,10,30,27\rbrace \\ &{}\lbrace 24,23,3,4,8,12,9,10,30,29\rbrace &{}\lbrace 24,21,1,4,8,12,11,10,30,31\rbrace &{}\lbrace 24,25,16,4,8,7,6,10,30,19\rbrace \\ &{}\lbrace 24,21,1,4,8,7,11,10,30,31\rbrace &{}\lbrace 24,23,3,4,8,5,9,10,30,29\rbrace &{}\lbrace 24,25,16,4,8,5,6,10,30,19\rbrace \\ &{}\lbrace 24,25,16,4,1,5,6,10,30,19\rbrace &{}\lbrace 24,23,3,4,1,5,9,10,30,29\rbrace &{}\lbrace 24,25,16,4,1,2,6,10,30,19\rbrace \\ &{}\lbrace 24,17,8,4,1,2,14,10,30,27\rbrace &{}\lbrace 24,23,3,4,1,13,9,10,30,29\rbrace &{}\lbrace 24,17,8,4,1,13,14,10,30,27\rbrace \\ &{}\lbrace 24,23,3,4,16,12,9,10,30,29\rbrace &{}\lbrace 24,21,1,4,16,12,11,10,30,31\rbrace &{}\lbrace 24,23,3,4,16,13,9,10,30,29 \rbrace \\ &{}\lbrace 24,17,8,4,16,13,14,10,30,27\rbrace &{}\lbrace 24,21,1,4,16,15,11,10,30,31\rbrace &{}\lbrace 24,17,8,4,16,15,14,10,30,27\rbrace \\ &{}\lbrace 17,32,12,8,4,3,2,14,27,22\rbrace &{}\lbrace 17,20,5,8,4,3,15,14,27,26\rbrace &{}\lbrace 17,18,7,8,4,1,13,14,27,28\rbrace \\ &{}\lbrace 17,32,12,8,4,1,2,14,27,22\rbrace &{}\lbrace 17,20,5,8,4,16,15,14,27,26\rbrace &{} \lbrace 17,18,7,8,4,16,13,14,27,28\rbrace \\ &{}\lbrace 17,32,12,8,5,1,2,14,27,22\rbrace &{}\lbrace 17,18,7,8,5,1,13,14,27,28\rbrace &{}\lbrace 17,32,12,8,5,6,2,14,27,22\rbrace \\ &{}\lbrace 17,24,4,8,5,6,10,14,27,30\rbrace &{}\lbrace 17,24,4,8,5,9,10,14,27,30\rbrace &{}\lbrace 17,18,7,8,5,9,13,14,27,28\rbrace \\ &{}\lbrace 17,24,4,8,7,6,10,14,27,30\rbrace &{}\lbrace 17,32,12,8,7,6,2,14,27,22\rbrace &{}\lbrace 17,24,4,8,7,11,10,14,27,30\rbrace \\ &{}\lbrace 17,20,5,8,7,11,15,14,27,26\rbrace &{}\lbrace 17,32,12,8,7,3,2,14,27,22\rbrace &{}\lbrace 17,20,5,8,7,3,15,14,27,26\rbrace \\ &{}\lbrace 17,24,4,8,12,11,10,14,27,30\rbrace &{}\lbrace 17,20,5,8,12,11,15,14,27,26\rbrace &{}\lbrace 17,18,7,8,12,9,13,14,27,28\rbrace \\ &{}\lbrace 17,24,4,8,12,9,10,14,27,30\rbrace &{}\lbrace 17,18,7,8,12,16,13,14,27,28\rbrace &{}\lbrace 17,20,5,8,12,16,15,14,27,26\rbrace \\ &{}\lbrace 31,32,12,11,7,3,2,1,21,22\rbrace &{}\lbrace 31,30,10,11,7,3,4,1,21,24\rbrace &{}\lbrace 31,32,12,11,7,6,2,1,21,22\rbrace \\ &{}\lbrace 31,26,15,11,7,6,5,1,21,20\rbrace &{}\lbrace 31,26,15,11,7,8,5,1,21,20\rbrace &{}\lbrace 31,30,10,11,7,8,4,1,21,24\rbrace \\ &{}\lbrace 31,26,15,11,10,6,5,1,21,20\rbrace &{}\lbrace 31,32,12,11,10,6,2,1,21,22\rbrace &{}\lbrace 31,26,15,11,10,9,5,1,21,20\rbrace \\ &{}\lbrace 31,18,7,11,10,9,13,1,21,28\rbrace &{}\lbrace 31,18,7,11,10,14,13,1,21,28\rbrace &{}\lbrace 31,32,12,11,10,14,2,1,21,22\rbrace \\ &{}\lbrace 31,26,15,11,12,8,5,1,21,20\rbrace &{}\lbrace 31,30,10,11,12,8,4,1,21,24\rbrace &{}\lbrace 31,18,7,11,12,9,13,1,21,28\rbrace \\ \end{array} \end{aligned}$$
$$\begin{aligned} \begin{array}{llll} &{}\lbrace 31,26,15,11,12,9,5,1,21,20\rbrace &{}\lbrace 31,18,7,11,12,16,13,1,21,28\rbrace &{}\lbrace 31,30,10,11,12,16,4,1,21,24\rbrace \\ &{}\lbrace 31,32,12,11,15,3,2,1,21,22\rbrace &{}\lbrace 31,30,10,11,15,3,4,1,21,24\rbrace &{}\lbrace 31,32,12,11,15,14,2,1,21,22 \rbrace \\ &{}\lbrace 31,18,7,11,15,14,13,1,21,28\rbrace &{}\lbrace 31,30,10,11,15,16,4,1,21,24\rbrace &{}\lbrace 31,18,7,11,15,16,13,1,21,28\rbrace \\ &{}\lbrace 26,31,11,15,3,2,1,5,20,21\rbrace &{}\lbrace 26,25,16,15,3,2,6,5,20,19\rbrace &{}\lbrace 26,31,11,15,3,4,1,5,20,21\rbrace \\ &{}\lbrace 26,27,14,15,3,4,8,5,20,17\rbrace &{}\lbrace 26,25,16,15,3,7,6,5,20,19\rbrace &{}\lbrace 26,27,14,15,3,7,8,5,20,17\rbrace \\ &{}\lbrace 26,25,16,15,11,7,6,5,20,19\rbrace &{}\lbrace 26,27,14,15,11,7,8,5,20,17\rbrace &{}\lbrace 26,23,3,15,11,10,9,5,20,29\rbrace \\ &{}\lbrace 26,25,16,15,11,10,6,5,20,19\rbrace &{}\lbrace 26,23,3,15,11,12,9,5,20,29\rbrace &{}\lbrace 26,27,14,15,11,12,8,5,20,17\rbrace \\ &{}\lbrace 26,31,11,15,14,2,1,5,20,21\rbrace &{}\lbrace 26,25,16,15,14,2,6,5,20,19\rbrace &{}\lbrace 26,23,3,15,14,10,9,5,20,29\rbrace \\ &{}\lbrace 26,25,16,15,14,10,6,5,20,19\rbrace &{}\lbrace 26,31,11,15,14,13,1,5,20,21\rbrace &{}\lbrace 26,23,3,15,14,13,9,5,20,29\rbrace \\ &{}\lbrace 26,31,11,15,16,4,1,5,20,21\rbrace &{}\lbrace 26,27,14,15,16,4,8,5,20,17\rbrace &{}\lbrace 26,23,3,15,16,13,9,5,20,29\rbrace \\ &{}\lbrace 26,31,11,15,16,13,1,5,20,21\rbrace &{}\lbrace 26,23,3,15,16,12,9,5,20,29\rbrace &{}\lbrace 26,27,14,15,16,12,8,5,20,17\rbrace \\ &{}\lbrace 23,26,15,3,2,1,5,9,29,20\rbrace &{}\lbrace 23,18,7,3,2,1,13,9,29,28\rbrace &{}\lbrace 23,26,15,3,2,6,5,9,29,20\rbrace \\ &{}\lbrace 23,24,4,3,2,6,10,9,29,30\rbrace &{}\lbrace 23,18,7,3,2,14,13,9,29,28\rbrace &{}\lbrace 23,24,4,3,2,14,10,9,29,30\rbrace \\ &{}\lbrace 23,26,15,3,4,1,5,9,29,20\rbrace &{}\lbrace 23,18,7,3,4,1,13,9,29,28\rbrace &{}\lbrace 23,26,15,3,4,8,5,9,29,20\rbrace \\ &{}\lbrace 23,22,2,3,4,8,12,9,29,32\rbrace &{}\lbrace 23,18,7,3,4,16,13,9,29,28\rbrace &{}\lbrace 23,22,2,3,4,16,12,9,29,32\rbrace \\ &{}\lbrace 23,26,15,3,7,6,5,9,29,20\rbrace &{}\lbrace 23,24,4,3,7,6,10,9,29,30\rbrace &{}\lbrace 23,26,15,3,7,8,5,9,29,20\rbrace \\ &{}\lbrace 23,22,2,3,7,8,12,9,29,32\rbrace &{}\lbrace 23,24,4,3,7,11,10,9,29,30\rbrace &{}\lbrace 23,22,2,3,7,11,12,9,29,32\rbrace \\ &{}\lbrace 23,24,4,3,15,11,10,9,29,30\rbrace &{}\lbrace 23,22,2,3,15,11,12,9,29,32\rbrace &{}\lbrace 23,18,7,3,15,14,13,9,29,28\rbrace \\ &{}\lbrace 23,24,4,3,15,14,10,9,29,30\rbrace &{}\lbrace 23,18,7,3,15,16,13,9,29,28\rbrace &{}\lbrace 23,22,2,3,15,16,12,9,29,32\rbrace \\ &{}\lbrace 18,31,11,7,3,2,1,13,28,21\rbrace &{}\lbrace 18,17,8,7,3,2,14,13,28,27\rbrace &{}\lbrace 18,31,11,7,3,4,1,13,28,21\rbrace \\ &{}\lbrace 18,19,6,7,3,4,16,13,28,25\rbrace &{}\lbrace 18,17,8,7,3,15,14,13,28,27\rbrace &{}\lbrace 18,19,6,7,3,15,16,13,28,25\rbrace \\ &{}\lbrace 18,17,8,7,6,2,14,13,28,27\rbrace &{}\lbrace 18,31,11,7,6,2,1,13,28,21\rbrace &{}\lbrace 18,31,11,7,6,5,1,13,28,21\rbrace \\ &{}\lbrace 18,23,3,7,6,5,9,13,28,29\rbrace &{}\lbrace 18,23,3,7,6,10,9,13,28,29\rbrace &{}\lbrace 18,17,8,7,6,10,14,13,28,27\rbrace \\ &{}\lbrace 18,31,11,7,8,4,1,13,28,21\rbrace &{}\lbrace 18,19,6,7,8,4,16,13,28,25\rbrace &{}\lbrace 18,31,11,7,8,5,1,13,28,21\rbrace \\ &{}\lbrace 18,23,3,7,8,5,9,13,28,29\rbrace &{}\lbrace 18,23,3,7,8,12,9,13,28,29\rbrace &{}\lbrace 18,19,6,7,8,12,16,13,28,25\rbrace \\ &{}\lbrace 18,23,3,7,11,10,9,13,28,29\rbrace &{}\lbrace 18,17,8,7,11,10,14,13,28,27\rbrace &{}\lbrace 18,23,3,7,11,12,9,13,28,29 \rbrace \\ &{}\lbrace 18,19,6,7,11,12,16,13,28,25\rbrace &{}\lbrace 18,17,8,7,11,15,14,13,28,27\rbrace &{}\lbrace 18,19,6,7,11,15,16,13,28,25\rbrace \\ &{}\lbrace 11,15,26,31,30,19,20,21,1,5\rbrace &{}\lbrace 11,12,32,31,30,19,22,21,1,2\rbrace &{}\lbrace 11,12,32,31,30,27,22,21,1,2\rbrace \\ &{}\lbrace 11,7,18,31,30,27,28,21,1,13\rbrace &{}\lbrace 11,15,26,31,30,29,20,21,1,5\rbrace &{}\lbrace 11,7,18,31,30,29,28,21,1,13\rbrace \\ &{}\lbrace 11,15,26,31,32,17,20,21,1,5\rbrace &{}\lbrace 11,10,30,31,32,17,24,21,1,4\rbrace &{}\lbrace 11,10,30,31,32,25,24,21,1,4\rbrace \\ &{}\lbrace 11,7,18,31,32,25,28,21,1,13\rbrace &{}\lbrace 11,15,26,31,32,29,20,21,1,5\rbrace &{}\lbrace 11,7,18,31,32,29,28,21,1,13\rbrace \\ &{}\lbrace 11,15,26,31,18,17,20,21,1,5\rbrace &{}\lbrace 11,10,30,31,18,17,24,21,1,4\rbrace &{}\lbrace 11,15,26,31,18,19,20,21,1,5\rbrace \\ &{}\lbrace 11,12,32,31,18,19,22,21,1,2\rbrace &{}\lbrace 11,10,30,31,18,23,24,21,1,4\rbrace &{}\lbrace 11,12,32,31,18,23,22,21,1,2\rbrace \\ &{}\lbrace 11,12,32,31,26,23,22,21,1,2\rbrace &{}\lbrace 11,10,30,31,26,23,24,21,1,4\rbrace &{}\lbrace 11,10,30,31,26,25,24,21,1,4\rbrace \\ &{}\lbrace 11,7,18,31,26,25,28,21,1,13\rbrace &{}\lbrace 11,7,18,31,26,27,28,21,1,13\rbrace &{}\lbrace 11,12,32,31,26,27,22,21,1,2\rbrace \\ &{}\lbrace 3,15,26,23,18,17,20,29,9,5\rbrace &{}\lbrace 3,2,22,23,18,17,32,29,9,12\rbrace &{}\lbrace 3,15,26,23,18,19,20,29,9,5\rbrace \\ &{}\lbrace 3,4,24,23,18,19,30,29,9,10\rbrace &{}\lbrace 3,4,24,23,18,31,30,29,9,10\rbrace &{}\lbrace 3,2,22,23,18,31,32,29,9,12\rbrace \\ &{}\lbrace 3,15,26,23,22,19,20,29,9,5\rbrace &{}\lbrace 3,4,24,23,22,19,30,29,9,10\rbrace &{}\lbrace 3,15,26,23,22,21,20,29,9,5\rbrace \\ &{}\lbrace 3,7,18,23,22,21,28,29,9,13\rbrace &{}\lbrace 3,7,18,23,22,27,28,29,9,13\rbrace &{}\lbrace 3,4,24,23,22,27,30,29,9,10\rbrace \\ &{}\lbrace 3,15,26,23,24,17,20,29,9,5\rbrace &{}\lbrace 3,2,22,23,24,17,32,29,9,12\rbrace &{}\lbrace 3,15,26,23,24,21,20,29,9,5\rbrace \\ &{}\lbrace 3,7,18,23,24,21,28,29,9,13\rbrace &{}\lbrace 3,7,18,23,24,25,28,29,9,13\rbrace &{}\lbrace 3,2,22,23,24,25,32,29,9,12\rbrace \\ &{}\lbrace 3,7,18,23,26,25,28,29,9,13\rbrace &{}\lbrace 3,2,22,23,26,25,32,29,9,12\rbrace &{}\lbrace 3,7,18,23,26,27,28,29,9,13\rbrace \\ &{}\lbrace 3,4,24,23,26,27,30,29,9,10\rbrace &{}\lbrace 3,4,24,23,26,31,30,29,9,10\rbrace &{}\lbrace 3,2,22,23,26,31,32,29,9,12\rbrace \\ &{}\lbrace 15,14,27,26,23,18,17,20,5,8\rbrace &{}\lbrace 15,16,25,26,23,18,19,20,5,6\rbrace &{}\lbrace 15,16,25,26,23,22,19,20,5,6\rbrace \\ &{}\lbrace 15,11,31,26,23,22,21,20,5,1\rbrace &{}\lbrace 15,11,31,26,23,24,21,20,5,1\rbrace &{}\lbrace 15,14,27,26,23,24,17,20,5,8\rbrace \\ &{}\lbrace 15,14,27,26,25,24,17,20,5,8\rbrace &{}\lbrace 15,11,31,26,25,24,21,20,5,1\rbrace &{}\lbrace 15,11,31,26,25,28,21,20,5,1\rbrace \\ &{}\lbrace 15,3,23,26,25,28,29,20,5,9\rbrace &{}\lbrace 15,3,23,26,25,32,29,20,5,9\rbrace &{}\lbrace 15,14,27,26,25,32,17,20,5,8 \rbrace \\ &{}\lbrace 15,16,25,26,27,22,19,20,5,6\rbrace &{}\lbrace 15,11,31,26,27,22,21,20,5,1\rbrace &{}\lbrace 15,11,31,26,27,28,21,20,5,1\rbrace \\ &{}\lbrace 15,3,23,26,27,28,29,20,5,9\rbrace &{}\lbrace 15,3,23,26,27,30,29,20,5,9\rbrace &{}\lbrace 15,16,25,26,27,30,19,20,5,6\rbrace \\ &{}\lbrace 15,14,27,26,31,18,17,20,5,8\rbrace &{}\lbrace 15,16,25,26,31,18,19,20,5,6\rbrace &{}\lbrace 15,16,25,26,31,30,19,20,5,6\rbrace \\ &{}\lbrace 15,3,23,26,31,30,29,20,5,9\rbrace &{}\lbrace 15,3,23,26,31,32,29,20,5,9\rbrace &{}\lbrace 15,14,27,26,31,32,17,20,5,8\rbrace \\ &{}\lbrace 4,16,25,24,17,18,19,30,10,6\rbrace &{}\lbrace 4,1,21,24,17,18,31,30,10,11\rbrace &{}\lbrace 4,3,23,24,17,20,29,30,10,9\rbrace \\ &{}\lbrace 4,16,25,24,17,20,19,30,10,6\rbrace &{}\lbrace 4,1,21,24,17,32,31,30,10,11\rbrace &{}\lbrace 4,3,23,24,17,32,29,30,10,9\rbrace \\ &{}\lbrace 4,3,23,24,21,20,29,30,10,9\rbrace &{}\lbrace 4,16,25,24,21,20,19,30,10,6\rbrace &{}\lbrace 4,8,17,24,21,22,27,30,10,14\rbrace \\ &{}\lbrace 4,16,25,24,21,22,19,30,10,6\rbrace &{}\lbrace 4,3,23,24,21,28,29,30,10,9\rbrace &{}\lbrace 4,8,17,24,21,28,27,30,10,14\rbrace \\ \end{array} \end{aligned}$$
$$\begin{aligned} \begin{array}{llll} &{}\lbrace 4,16,25,24,23,18,19,30,10,6\rbrace &{}\lbrace 4,1,21,24,23,18,31,30,10,11\rbrace &{}\lbrace 4,16,25,24,23,22,19,30,10,6\rbrace \\ &{}\lbrace 4,8,17,24,23,22,27,30,10,14\rbrace &{}\lbrace 4,8,17,24,23,26,27,30,10,14\rbrace &{}\lbrace 4,1,21,24,23,26,31,30,10,11\rbrace \\ &{}\lbrace 4,8,17,24,25,26,27,30,10,14,\rbrace &{}\lbrace 4,1,21,24,25,26,31,30,10,11\rbrace &{}\lbrace 4,8,17,24,25,28,27,30,10,14\rbrace \\ &{}\lbrace 4,3,23,24,25,28,29,30,10,9\rbrace &{}\lbrace 4,3,23,24,25,32,29,30,10,9\rbrace &{}\lbrace 4,1,21,24,25,32,31,30,10,11\rbrace \\ &{}\lbrace 12,9,29,32,25,26,23,22,2,3\rbrace &{}\lbrace 12,8,17,32,25,26,27,22,2,14\rbrace &{}\lbrace 12,11,31,32,25,28,21,22,2,1\rbrace \\ &{}\lbrace 12,8,17,32,25,28,27,22,2,14\rbrace &{}\lbrace 12,9,29,32,25,24,23,22,2,3\rbrace &{}\lbrace 12,11,31,32,25,24,21,22,2,1\rbrace \\ &{}\lbrace 12,16,25,32,17,18,19,22,2,6\rbrace &{}\lbrace 12,9,29,32,17,18,23,22,2,3\rbrace &{}\lbrace 12,16,25,32,17,20,19,22,2,6\rbrace \\ &{}\lbrace 12,11,31,32,17,20,21,22,2,1\rbrace &{}\lbrace 12,9,29,32,17,24,23,22,2,3\rbrace &{}\lbrace 12,11,31,32,17,24,21,22,2,1\rbrace \\ &{}\lbrace 12,16,25,32,31,18,19,22,2,6\rbrace &{}\lbrace 12,9,29,32,31,18,23,22,2,3\rbrace &{}\lbrace 12,16,25,32,31,30,19,22,2,6\rbrace \\ &{}\lbrace 12,8,17,32,31,30,27,22,2,14\rbrace &{}\lbrace 12,9,29,32,31,26,23,22,2,3\rbrace &{}\lbrace 12,8,17,32,31,26,27,22,2,14\rbrace \\ &{}\lbrace 12,11,31,32,29,20,21,22,2,1\rbrace &{}\lbrace 12,16,25,32,29,20,19,22,2,6\rbrace &{}\lbrace 12,11,31,32,29,28,21,22,2,1\rbrace \\ &{}\lbrace 12,8,17,32,29,28,27,22,2,14\rbrace &{}\lbrace 12,16,25,32,29,30,19,22,2,6\rbrace &{}\lbrace 12,8,17,32,29,30,27,22,2,14\rbrace \\ &{}\lbrace 8,12,32,17,18,19,22,27,14,2\rbrace &{}\lbrace 8,4,24,17,18,19,30,27,14,10\rbrace &{}\lbrace 8,12,32,17,18,23,22,27,14,2\rbrace \\ &{}\lbrace 8,5,20,17,18,23,26,27,14,15\rbrace &{}\lbrace 8,5,20,17,18,31,26,27,14,15\rbrace &{}\lbrace 8,4,24,17,18,31,30,27,14,10\rbrace \\ &{}\lbrace 8,12,32,17,20,19,22,27,14,2\rbrace &{}\lbrace 8,4,24,17,20,19,30,27,14,10\rbrace &{}\lbrace 8,12,32,17,20,21,22,27,14,2\rbrace \\ &{}\lbrace 8,7,18,17,20,21,28,27,14,13\rbrace &{}\lbrace 8,7,18,17,20,29,28,27,14,13\rbrace &{}\lbrace 8,4,24,17,20,29,30,27,14,10\rbrace \\ &{}\lbrace 8,12,32,17,24,21,22,27,14,2\rbrace &{}\lbrace 8,7,18,17,24,21,28,27,14,13\rbrace &{}\lbrace 8,12,32,17,24,23,22,27,14,2\rbrace \\ &{}\lbrace 8,5,20,17,24,23,26,27,14,15\rbrace &{}\lbrace 8,5,20,17,24,25,26,27,14,15\rbrace &{}\lbrace 8,7,18,17,24,25,28,27,14,13\rbrace \\ &{}\lbrace 8,5,20,17,32,25,26,27,14,15\rbrace &{}\lbrace 8,7,18,17,32,25,28,27,14,13\rbrace &{}\lbrace 8,7,18,17,32,29,28,27,14,13\rbrace \\ &{}\lbrace 8,4,24,17,32,29,30,27,14,10\rbrace &{}\lbrace 8,5,20,17,32,31,26,27,14,15\rbrace &{}\lbrace 8,4,24,17,32,31,30,27,14,10\rbrace \\ &{}\lbrace 16,13,28,25,24,17,18,19,6,7\rbrace &{}\lbrace 16,15,26,25,24,17,20,19,6,5\rbrace &{}\lbrace 16,15,26,25,24,21,20,19,6,5\rbrace \\ &{}\lbrace 16,12,32,25,24,21,22,19,6,2\rbrace &{}\lbrace 16,13,28,25,24,23,18,19,6,7\rbrace &{}\lbrace 16,12,32,25,24,23,22,19,6,2\rbrace \\ &{}\lbrace 16,13,28,25,26,23,18,19,6,7\rbrace &{}\lbrace 16,12,32,25,26,23,22,19,6,2\rbrace &{}\lbrace 16,12,32,25,26,27,22,19,6,2\rbrace \\ &{}\lbrace 16,4,24,25,26,27,30,19,6,10\rbrace &{}\lbrace 16,13,28,25,26,31,18,19,6,7\rbrace &{}\lbrace 16,4,24,25,26,31,30,19,6,10\rbrace \\ &{}\lbrace 16,15,26,25,28,21,20,19,6,5\rbrace &{}\lbrace 16,12,32,25,28,21,22,19,6,2\rbrace &{}\lbrace 16,12,32,25,28,27,22,19,6,2\rbrace \\ &{}\lbrace 16,4,24,25,28,27,30,19,6,10\rbrace &{}\lbrace 16,15,26,25,28,29,20,19,6,5\rbrace &{}\lbrace 16,4,24,25,28,29,30,19,6,10\rbrace \\ &{}\lbrace 16,13,28,25,32,17,18,19,6,7\rbrace &{}\lbrace 16,15,26,25,32,17,20,19,6,5\rbrace &{}\lbrace 16,15,26,25,32,29,20,19,6,5\rbrace \\ &{}\lbrace 16,4,24,25,32,29,30,19,6,10\rbrace &{}\lbrace 16,13,28,25,32,31,18,19,6,7\rbrace &{}\lbrace 16,4,24,25,32,31,30,19,6,10\rbrace \\ &{}\lbrace 7,11,31,18,17,24,21,28,13,1\rbrace &{}\lbrace 7,6,19,18,17,24,25,28,13,16\rbrace &{}\lbrace 7,11,31,18,17,20,21,28,13,1\rbrace \\ &{}\lbrace 7,3,23,18,17,20,29,28,13,9\rbrace &{}\lbrace 7,6,19,18,17,32,25,28,13,16\rbrace &{}\lbrace 7,3,23,18,17,32,29,28,13,9\rbrace \\ &{}\lbrace 7,11,31,18,19,20,21,28,13,1\rbrace &{}\lbrace 7,3,23,18,19,20,29,28,13,9\rbrace &{}\lbrace 7,11,31,18,19,22,21,28,13,1\rbrace \\ &{}\lbrace 7,8,17,18,19,22,27,28,13,14\rbrace &{}\lbrace 7,8,17,18,19,30,27,28,13,14\rbrace &{}\lbrace 7,3,23,18,19,30,29,28,13,9\rbrace \\ &{}\lbrace 7,11,31,18,23,22,21,28,13,1\rbrace &{}\lbrace 7,8,17,18,23,22,27,28,13,14\rbrace &{}\lbrace 7,11,31,18,23,24,21,28,13,1\rbrace \\ &{}\lbrace 7,6,19,18,23,26,25,28,13,16\rbrace &{}\lbrace 7,6,19,18,23,26,25,28,13,16\rbrace &{}\lbrace 7,8,17,18,23,26,27,28,13,14\rbrace \\ &{}\lbrace 7,6,19,18,31,26,25,28,13,16\rbrace &{}\lbrace 7,8,17,18,31,26,27,28,13,14\rbrace &{}\lbrace 7,8,17,18,31,30,27,28,13,14\rbrace \\ &{}\lbrace 7,3,23,18,31,30,29,28,13,9\rbrace &{}\lbrace 7,3,23,18,31,32,29,28,13,9\rbrace &{}\lbrace 7,6,19,18,31,32,25,28,13,16\rbrace \\ &{}\lbrace 22,27,14,2,3,7,8,12,32,17\rbrace &{}\lbrace 22,21,1,2,3,7,11,12,32,31\rbrace &{}\lbrace 30,29,9,10,6,7,3,4,24,23\rbrace \\ &{}\lbrace 22,21,1,2,3,15,11,12,32,31\rbrace &{}\lbrace 22,19,6,2,3,15,16,12,32,25\rbrace &{}\lbrace 30,27,14,10,6,7,8,4,24,17\rbrace \\ &{}\lbrace 22,19,6,2,3,4,16,12,32,25\rbrace &{}\lbrace 22,27,14,2,3,4,8,12,32,17\rbrace &{}\lbrace 30,31,11,10,6,5,1,4,24,21\rbrace \\ &{}\lbrace 22,23,3,2,1,5,9,12,32,29\rbrace &{}\lbrace 22,27,14,2,1,5,8,12,32,17\rbrace &{}\lbrace 30,27,14,10,6,5,8,4,24,17\rbrace \\ &{}\lbrace 22,19,6,2,1,4,16,12,32,25\rbrace &{}\lbrace 22,27,14,2,1,4,8,12,32,17\rbrace &{}\lbrace 30,29,9,10,6,2,3,4,24,23\rbrace \\ &{}\lbrace 22,23,3,2,1,13,9,12,32,29\rbrace &{}\lbrace 22,19,6,2,1,13,16,12,32,25\rbrace &{}\lbrace 30,31,11,10,6,2,1,4,24,21\rbrace \\ &{}\lbrace 22,23,3,2,6,5,9,12,32,29\rbrace &{}\lbrace 22,27,14,2,6,5,8,12,32,17 \rbrace &{}\lbrace 30,31,11,10,14,13,1,4,24,21\rbrace \\ &{}\lbrace 22,27,14,2,6,7,8,12,32,17\rbrace &{}\lbrace 22,21,1,2,6,7,11,12,32,31\rbrace &{}\lbrace 30,19,6,10,14,13,16,4,24,25\rbrace \\ &{}\lbrace 22,21,1,2,6,10,11,12,32,31\rbrace &{}\lbrace 22,23,3,2,6,10,9,12,32,29\rbrace &{}\lbrace 30,19,6,10,14,15,16,4,24,25 \rbrace \\ &{}\lbrace 22,21,1,2,14,10,11,12,32,31\rbrace &{}\lbrace 22,23,3,2,14,10,9,12,32,29\rbrace &{}\lbrace 30,29,9,10,14,15,3,4,24,23\rbrace \\ &{}\lbrace 22,23,3,2,14,13,9,12,32,29\rbrace &{}\lbrace 22,19,6,2,14,13,16,12,32,25\rbrace &{}\lbrace 30,29,9,10,14,2,3,4,24,23\rbrace \\ &{}\lbrace 22,19,6,2,14,15,16,12,32,25\rbrace &{}\lbrace 22,21,1,2,14,15,11,12,32,31 \rbrace &{}\lbrace 30,31,11,10,14,2,1,4,24,21 \rbrace \\ &{}\lbrace 19,30,10,6,5,1,4,16,25,24\rbrace &{}\lbrace 19,18,7,6,5,1,13,16,25,28\rbrace &{}\lbrace 30,29,9,10,11,7,3,4,24,23 \rbrace \\ &{}\lbrace 19,18,7,6,5,9,13,16,25,28\rbrace &{}\lbrace 19,22,2,6,5,9,12,16,25,32\rbrace &{}\lbrace 30,27,14,10,11,7,8,4,24,17 \rbrace \\ &{}\lbrace 19,30,10,6,5,8,4,16,25,24\rbrace &{}\lbrace 19,22,2,6,5,8,12,16,25,32\rbrace &{}\lbrace 30,27,14,10,11,12,8,4,24,17 \rbrace \\ &{}\lbrace 19,30,10,6,7,8,4,16,25,24\rbrace &{}\lbrace 19,22,2,6,7,8,12,16,25,32\rbrace &{}\lbrace 30,19,6,10,11,12,16,4,24,25 \rbrace \\ &{}\lbrace 19,22,2,6,7,11,12,16,25,32\rbrace &{}\lbrace 19,20,5,6,7,11,15,16,25,26\rbrace &{}\lbrace 30,19,6,10,11,15,16,4,24,25 \rbrace \\ &{}\lbrace 19,30,10,6,7,3,4,16,25,24\rbrace &{}\lbrace 19,20,5,6,7,3,15,16,25,26\rbrace &{}\lbrace 30,29,9,10,11,15,3,4,24,23 \rbrace \\ \end{array} \end{aligned}$$
$$\begin{aligned} \begin{array}{llll} &{}\lbrace 19,22,2,6,10,9,12,16,25,32\rbrace &{}\lbrace 19,18,7,6,10,9,13,16,25,28\rbrace &{}\lbrace 30,31,11,10,9,5,1,4,24,21\rbrace \\ &{}\lbrace 19,22,2,6,10,11,12,16,25,32\rbrace &{}\lbrace 19,20,5,6,10,11,15,16,25,26\rbrace &{}\lbrace 30,27,14,10,9,5,8,4,24,17 \rbrace \\ &{}\lbrace 19,20,5,6,10,14,15,16,25,26\rbrace &{}\lbrace 19,18,7,6,10,14,13,16,25,28\rbrace &{}\lbrace 30,31,11,10,9,13,1,4,24,21 \rbrace \\ &{}\lbrace 19,30,10,6,2,1,4,16,25,24\rbrace &{}\lbrace 19,18,7,6,2,1,13,16,25,28\rbrace &{}\lbrace 30,19,6,10,9,13,16,4,24,25 \rbrace \\ &{}\lbrace 19,30,10,6,2,3,4,16,25,24\rbrace &{}\lbrace 19,20,5,6,2,3,15,16,25,26\rbrace &{}\lbrace 30,27,14,10,9,12,8,4,24,17 \rbrace \\ &{}\lbrace 19,20,5,6,2,14,15,16,25,26\rbrace &{}\lbrace 19,18,7,6,2,14,13,16,25,28\rbrace &{}\lbrace 30,19,6,10,9,12,16,4,24,25 \rbrace \\ &{}\lbrace 27,26,15,14,13,9,5,8,17,20\rbrace &{}\lbrace 27,22,2,14,13,9,12,8,17,32\rbrace &{}\lbrace 27,26,15,14,13,1,5,8,17,20\rbrace \\ &{}\lbrace 27,30,10,14,13,1,4,8,17,24\rbrace &{}\lbrace 27,30,10,14,13,16,4,8,17,24\rbrace &{}\lbrace 27,22,2,14,13,16,12,8,17,32\rbrace \\ &{}\lbrace 27,28,13,14,15,11,7,8,17,18\rbrace &{}\lbrace 27,22,2,14,15,11,12,8,17,32\rbrace &{}\lbrace 27,28,13,14,15,3,7,8,17,18\rbrace \\ &{}\lbrace 27,30,10,14,15,3,4,8,17,24\rbrace &{}\lbrace 27,30,10,14,15,16,4,8,17,24\rbrace &{}\lbrace 27,22,2,14,15,16,12,8,17,32\rbrace \\ &{}\lbrace 27,26,15,14,10,9,5,8,17,20\rbrace &{}\lbrace 27,22,2,14,10,9,12,8,17,32\rbrace &{}\lbrace 27,28,13,14,10,11,7,8,17,18\rbrace \\ &{}\lbrace 27,22,2,14,10,11,12,8,17,32\rbrace &{}\lbrace 27,28,13,14,2,3,7,8,17,18\rbrace &{}\lbrace 27,30,10,14,2,3,4,8,17,24\rbrace \\ &{}\lbrace 27,28,13,14,,10,6,7,8,17,18\rbrace &{}\lbrace 27,26,15,14,10,6,5,8,17,20\rbrace &{}\lbrace 27,26,15,14,2,1,5,8,17,20\rbrace \\ &{}\lbrace 27,30,10,14,2,1,4,8,17,24\rbrace &{}\lbrace 27,28,13,14,2,6,7,8,17,18\rbrace &{}\lbrace 27,26,15,14,2,6,5,8,17,20\rbrace \\ &{}\lbrace 21,28,13,1,2,6,7,11,31,18\rbrace &{}\lbrace 21,24,4,1,2,6,10,11,31,30\rbrace &{}\lbrace 21,28,13,1,2,3,7,11,31,18\rbrace \\ &{}\lbrace 21,20,5,1,2,3,15,11,31,26\rbrace &{}\lbrace 21,24,4,1,2,14,10,11,31,30\rbrace &{}\lbrace 21,20,5,1,2,14,15,11,31,26\rbrace \\ &{}\lbrace 21,28,13,1,4,3,7,11,31,18\rbrace &{}\lbrace 21,20,5,1,4,3,15,11,31,26\rbrace &{}\lbrace 21,22,2,1,4,8,12,11,31,32\rbrace \\ &{}\lbrace 21,28,13,1,4,8,7,11,31,18\rbrace &{}\lbrace 21,20,5,1,4,16,15,11,31,26\rbrace &{}\lbrace 21,22,2,1,4,16,12,11,31,32\rbrace \\ &{}\lbrace 21,28,13,1,5,6,7,11,31,18\rbrace &{}\lbrace 21,24,4,1,5,6,10,11,31,30\rbrace &{}\lbrace 21,24,4,1,5,9,10,11,31,30\rbrace \\ &{}\lbrace 21,22,2,1,5,9,12,11,31,32\rbrace &{}\lbrace 21,28,13,1,5,8,7,11,31,18\rbrace &{}\lbrace 21,22,2,1,5,8,12,11,31,32\rbrace \\ &{}\lbrace 21,24,4,1,13,9,10,11,31,30\rbrace &{}\lbrace 21,22,2,1,13,9,12,11,31,32\rbrace &{}\lbrace 21,24,4,1,13,14,10,11,31,30 \rbrace \\ &{}\lbrace 21,20,5,1,13,14,15,11,31,26\rbrace &{}\lbrace 21,20,5,1,13,16,15,11,31,26\rbrace &{}\lbrace 21,22,2,1,13,16,12,11,31,32\rbrace \\ &{}\lbrace 13,1,21,28,27,26,31,18,7,11\rbrace &{}\lbrace 13,16,25,28,27,30,19,18,7,6\rbrace &{}\lbrace 13,1,21,28,27,30,31,18,7,11\rbrace \\ &{}\lbrace 13,17,27,28,29,20,17,18,7,8\rbrace &{}\lbrace 13,16,25,28,29,20,19,18,7,6\rbrace &{}\lbrace 13,16,25,28,29,30,19,18,7,6\rbrace \\ &{}\lbrace 13,1,21,28,29,30,31,18,7,11\rbrace &{}\lbrace 13,1,21,28,29,32,31,18,7,11\rbrace &{}\lbrace 13,14,27,28,29,32,17,18,7,8\rbrace \\ &{}\lbrace 20,29,19,5,1,2,3,15,26,23\rbrace &{}\lbrace 20,17,8,5,1,2,14,15,26,27\rbrace &{}\lbrace 20,29,9,5,1,4,3,15,26,23\rbrace \\ &{}\lbrace 20,19,6,5,1,4,16,15,26,25\rbrace &{}\lbrace 20,17,8,5,1,13,14,15,26,27\rbrace &{}\lbrace 20,19,6,5,1,13,16,15,26,25\rbrace \\ &{}\lbrace 20,21,1,5,6,7,11,15,26,31\rbrace &{}\lbrace 20,29,9,5,6,7,3,15,26,23\rbrace &{}\lbrace 20,29,9,5,6,2,3,15,26,23\rbrace \\ &{}\lbrace 20,17,8,5,6,2,14,15,26,27\rbrace &{}\lbrace 20,21,1,5,6,10,11,15,26,31\rbrace &{}\lbrace 20,17,8,5,6,10,14,15,26,27\rbrace \\ &{}\lbrace 20,21,1,5,8,7,11,15,26,31\rbrace &{}\lbrace 20,29,9,5,8,7,3,15,26,23\rbrace &{}\lbrace 20,29,9,5,8,4,3,15,26,23\rbrace \\ &{}\lbrace 20,19,6,5,8,4,16,15,26,25\rbrace &{}\lbrace 20,19,6,5,8,12,16,15,26,25\rbrace &{}\lbrace 20,21,1,5,8,12,11,15,26,31\rbrace \\ &{}\lbrace 20,21,1,5,9,10,11,15,26,31\rbrace &{}\lbrace 20,17,8,5,9,10,14,15,26,27\rbrace &{}\lbrace 20,17,8,5,9,13,14,15,26,27\rbrace \\ &{}\lbrace 20,19,6,5,9,13,16,15,26,25\rbrace &{}\lbrace 20,19,6,5,9,12,16,15,26,25\rbrace &{}\lbrace 20,21,1,5,9,12,11,15,26,31\rbrace \\ &{}\lbrace 29,32,12,9,5,6,2,3,23,22\rbrace &{}\lbrace 29,28,13,9,5,6,7,3,23,18\rbrace &{}\lbrace 29,32,12,9,5,1,2,3,23,22\rbrace \\ &{}\lbrace 29,30,10,9,5,1,4,3,23,24\rbrace &{}\lbrace 29,30,10,9,5,8,4,3,23,24\rbrace &{}\lbrace 29,28,13,9,5,8,7,3,23,18\rbrace \\ &{}\lbrace 29,32,12,9,10,6,2,3,23,22\rbrace &{}\lbrace 29,28,13,9,10,6,7,3,23,18\rbrace &{}\lbrace 29,20,5,9,10,11,15,3,23,26\rbrace \\ &{}\lbrace 29,32,12,9,10,14,2,3,23,22\rbrace &{}\lbrace 29,20,5,9,10,14,15,3,23,26\rbrace &{}\lbrace 29,28,13,9,12,11,7,3,23,18\rbrace \\ &{}\lbrace 29,20,5,9,12,11,15,3,23,26\rbrace &{}\lbrace 29,30,10,9,12,8,4,3,23,24\rbrace &{}\lbrace 29,28,13,9,12,8,7,3,23,18\rbrace \\ &{}\lbrace 29,30,10,9,12,16,4,3,23,24\rbrace &{}\lbrace 29,20,5,9,12,16,15,3,23,26\rbrace &{}\lbrace 29,32,12,9,13,1,2,3,23,22\rbrace \\ &{}\lbrace 29,30,10,9,13,1,4,3,23,24\rbrace &{}\lbrace 29,20,5,9,13,14,15,3,23,26\rbrace &{}\lbrace 29,32,12,9,13,14,2,3,23,22\rbrace \\ &{}\lbrace 29,30,10,9,13,16,4,3,23,24\rbrace &{}\lbrace 29,20,5,9,13,16,15,3,23,26\rbrace &{}\lbrace 29,28,13,9,10,11,7,3,23,18\rbrace \\ &{}\lbrace 28,25,16,13,1,5,6,7,18,19\rbrace &{}\lbrace 28,27,14,13,1,5,8,7,18,17\rbrace &{}\lbrace 28,25,16,13,1,2,6,7,18,19\rbrace \\ &{}\lbrace 28,29,9,13,1,2,3,7,18,23\rbrace &{}\lbrace 28,29,9,13,1,4,3,7,18,23\rbrace &{}\lbrace 28,27,14,13,9,12,8,7,18,17\rbrace \\ &{}\lbrace 28,25,16,13,9,5,6,7,18,19\rbrace &{}\lbrace 28,27,14,13,9,5,8,7,18,17\rbrace &{}\lbrace 28,25,16,13,9,10,6,7,18,19\rbrace \\ &{}\lbrace 28,21,1,13,9,10,11,7,18,31\rbrace &{}\lbrace 28,21,1,13,9,12,11,7,18,31\rbrace &{}\lbrace 28,27,14,13,9,12,8,7,18,17\rbrace \\ &{}\lbrace 28,25,16,13,14,2,6,7,18,19\rbrace &{}\lbrace 28,29,9,13,14,2,3,7,18,23\rbrace &{}\lbrace 28,25,16,13,14,10,6,7,18,19\rbrace \\ &{}\lbrace 28,21,1,13,14,10,11,7,18,31\rbrace &{}\lbrace 28,21,1,13,14,15,11,7,18,31\rbrace &{}\lbrace 28,29,9,13,14,15,3,7,18,23\rbrace \\ &{}\lbrace 28,29,9,13,16,4,3,7,18,23\rbrace &{}\lbrace 28,27,14,13,16,4,8,7,18,17\rbrace &{}\lbrace 28,27,14,13,16,12,8,7,18,17\rbrace \\ &{}\lbrace 28,21,1,13,16,12,11,7,18,31\rbrace &{}\lbrace 28,21,1,13,16,15,11,7,18,31\rbrace &{}\lbrace 28,29,9,13,16,15,3,7,18,23\rbrace \\ &{}\lbrace 1,13,28,21,20,17,18,31,11,7\rbrace &{}\lbrace 1,2,22,21,20,17,32,31,11,12\rbrace &{}\lbrace 1,13,28,21,20,19,18,31,11,7\rbrace \\ &{}\lbrace 1,4,24,21,20,19,30,31,11,10\rbrace &{}\lbrace 1,4,24,21,20,29,30,31,11,10\rbrace &{}\lbrace 1,2,22,21,20,29,32,31,11,12\rbrace \\ &{}\lbrace 1,13,28,21,22,19,18,31,11,7\rbrace &{}\lbrace 1,4,24,21,22,19,30,31,11,10\rbrace &{}\lbrace 1,13,28,21,22,23,18,31,11,7\rbrace \\ \end{array} \end{aligned}$$
$$\begin{aligned} \begin{array}{llll} &{}\lbrace 1,5,20,21,22,23,26,31,11,15\rbrace &{}\lbrace 1,5,20,21,22,27,26,31,11,15\rbrace &{}\lbrace 1,4,24,21,22,27,30,31,11,10\rbrace \\ &{}\lbrace 1,13,28,21,24,17,18,31,11,7\rbrace &{}\lbrace 1,2,22,21,24,17,32,31,11,12\rbrace &{}\lbrace 1,13,28,21,24,23,18,31,11,7\rbrace \\ &{}\lbrace 1,5,20,21,24,23,26,31,11,15\rbrace &{}\lbrace 1,5,20,21,24,25,26,31,11,15\rbrace &{}\lbrace 1,2,22,21,24,25,32,31,11,12\rbrace \\ &{}\lbrace 1,5,20,21,28,25,26,31,11,15\rbrace &{}\lbrace 1,2,22,21,28,25,32,31,11,12\rbrace &{}\lbrace 1,5,20,21,28,27,26,31,11,15\rbrace \\ &{}\lbrace 1,4,24,21,28,27,30,31,11,10\rbrace &{}\lbrace 1,4,24,21,28,29,30,31,11,10\rbrace &{}\lbrace 1,2,22,21,28,29,32,31,11,12\rbrace \\ &{}\lbrace 9,13,28,29,20,17,18,23,3,7\rbrace &{}\lbrace 9,10,30,29,20,17,24,23,3,4\rbrace &{}\lbrace 9,13,28,29,20,19,18,23,3,7\rbrace \\ &{}\lbrace 9,12,32,29,20,19,22,23,3,2\rbrace &{}\lbrace 9,12,32,29,20,21,22,23,3,2\rbrace &{}\lbrace 9,10,30,29,20,21,24,23,3,4\rbrace \\ &{}\lbrace 9,13,28,29,30,19,18,23,3,7\rbrace &{}\lbrace 9,12,32,29,30,19,22,23,3,2\rbrace &{}\lbrace 9,12,32,29,30,27,22,23,3,2\rbrace \\ &{}\lbrace 9,5,20,29,30,27,26,23,3,15\rbrace &{}\lbrace 9,5,20,29,30,31,26,23,3,15\rbrace &{}\lbrace 9,13,28,29,30,31,18,23,3,7\rbrace \\ &{}\lbrace 9,13,28,29,32,17,18,23,3,7\rbrace &{}\lbrace 9,10,30,29,32,17,24,23,3,4\rbrace &{}\lbrace 9,10,30,29,32,25,24,23,3,4\rbrace \\ &{}\lbrace 9,5,20,29,32,25,26,23,3,15\rbrace &{}\lbrace 9,5,20,29,32,31,26,23,3,15\rbrace &{}\lbrace 9,13,28,29,32,31,18,23,3,7\rbrace \\ &{}\lbrace 9,12,32,29,28,21,22,23,3,2\rbrace &{}\lbrace 9,10,30,29,28,21,24,23,3,4\rbrace &{}\lbrace 9,12,32,29,28,27,22,23,3,2\rbrace \\ &{}\lbrace 9,5,20,29,28,27,26,23,3,15\rbrace &{}\lbrace 9,10,30,29,28,25,24,23,3,4\rbrace &{}\lbrace 9,5,20,29,28,25,26,23,3,15\rbrace \\ &{}\lbrace 5,9,29,20,17,18,23,26,15,3\rbrace &{}\lbrace 5,1,21,20,17,18,31,26,15,11\rbrace &{}\lbrace 5,9,29,20,17,24,23,26,15,3 \rbrace \\ &{}\lbrace 5,6,19,20,17,24,25,26,15,16\rbrace &{}\lbrace 5,6,19,20,17,32,25,26,15,16\rbrace &{}\lbrace 5,1,21,20,17,32,31,26,15,11\rbrace \\ &{}\lbrace 5,9,29,20,19,18,23,26,15,3\rbrace &{}\lbrace 5,1,21,20,19,18,31,26,15,11\rbrace &{}\lbrace 5,9,29,20,19,22,23,26,15,3 \rbrace \\ &{}\lbrace 5,8,17,20,19,22,27,26,15,14\rbrace &{}\lbrace 5,8,17,20,19,30,27,26,15,14\rbrace &{}\lbrace 5,1,21,22,19,30,31,26,15,11\rbrace \\ &{}\lbrace 5,9,29,20,21,22,23,26,15,3\rbrace &{}\lbrace 5,8,17,20,21,22,27,26,15,14\rbrace &{}\lbrace 5,9,29,20,21,24,23,26,15,3 \rbrace \\ &{}\lbrace 5,6,19,20,21,24,25,26,15,16\rbrace &{}\lbrace 5,6,19,20,21,28,25,26,15,16\rbrace &{}\lbrace 5,8,17,20,21,28,27,26,15,14\rbrace \\ &{}\lbrace 5,6,19,20,29,28,25,26,15,16\rbrace &{}\lbrace 5,8,17,20,29,28,27,26,15,14\rbrace &{}\lbrace 5,8,17,20,29,30,27,26,15,14\rbrace \\ &{}\lbrace 5,1,21,20,29,30,31,26,15,11\rbrace &{}\lbrace 5,6,19,20,29,32,25,26,15,16\rbrace &{}\lbrace 5,1,21,20,29,32,31,26,15,11\rbrace \\ &{}\lbrace 10,11,31,30,19,20,21,24,4,1\rbrace &{}\lbrace 10,14,27,30,19,20,17,24,4,8\rbrace &{}\lbrace 10,14,27,30,19,18,17,24,4,8\rbrace \\ &{}\lbrace 10,9,29,30,19,18,23,24,4,3\rbrace &{}\lbrace 10,9,29,30,19,22,23,24,4,3\rbrace &{}\lbrace 10,11,31,30,19,22,21,24,4,1 \rbrace \\ &{}\lbrace 10,6,19,30,31,32,25,24,4,16\rbrace &{}\lbrace 10,14,27,30,31,32,17,24,4,8\rbrace &{}\lbrace 10,14,27,30,31,18,17,24,4,8\rbrace \\ &{}\lbrace 10,9,29,30,31,18,23,24,4,3\rbrace &{}\lbrace 10,9,29,30,31,26,23,24,4,3\rbrace &{}\lbrace 10,6,19,30,31,26,25,24,4,16 \rbrace \\ &{}\lbrace 10,11,31,30,29,20,21,24,4,1\rbrace &{}\lbrace 10,14,27,30,29,20,17,24,4,8\rbrace &{}\lbrace 10,11,31,30,29,28,21,24,4,1\rbrace \\ &{}\lbrace 10,6,19,30,29,28,25,24,4,16\rbrace &{}\lbrace 10,14,27,30,29,32,17,24,4,8\rbrace &{}\lbrace 10,6,19,30,29,32,25,24,4,16\rbrace \\ &{}\lbrace 10,11,31,30,27,28,21,24,4,1\rbrace &{}\lbrace 10,6,19,30,27,28,25,24,4,16\rbrace &{}\lbrace 10,9,29,30,27,26,23,24,4,3\rbrace \\ &{}\lbrace 10,6,19,30,27,26,25,24,4,16\rbrace &{}\lbrace 10,9,29,30,27,22,23,24,4,3\rbrace &{}\lbrace 10,11,31,30,27,22,21,24,4,1\rbrace \\ &{}\lbrace 2,3,23,22,21,20,29,32,12,9\rbrace &{}\lbrace 2,14,27,22,21,20,17,32,12,8\rbrace &{}\lbrace 2,3,23,22,21,28,29,32,12,9 \rbrace \\ &{}\lbrace 2,6,19,22,21,28,25,32,12,16\rbrace &{}\lbrace 2,6,19,22,21,24,25,32,12,16\rbrace &{}\lbrace 2,14,27,22,21,24,17,32,12,8\rbrace \\ &{}\lbrace 2,14,27,22,23,18,17,32,12,8\rbrace &{}\lbrace 2,1,21,22,23,18,31,32,12,11\rbrace &{}\lbrace 2,6,19,22,23,24,25,32,12,16\rbrace \\ &{}\lbrace 2,14,27,22,23,24,17,32,12,8\rbrace &{}\lbrace 2,6,19,22,23,26,25,32,12,16\rbrace &{}\lbrace 2,1,21,22,23,26,31,32,12,11\rbrace \\ &{}\lbrace 2,14,27,22,19,18,17,32,12,8\rbrace &{}\lbrace 2,1,21,22,19,18,31,32,12,11\rbrace &{}\lbrace 2,3,23,22,19,20,29,32,12,9\rbrace \\ &{}\lbrace 2,14,27,22,19,20,17,32,12,8\rbrace &{}\lbrace 2,1,21,22,19,30,31,32,12,11\rbrace &{}\lbrace 2,3,23,22,19,30,29,32,12,9\rbrace \\ &{}\lbrace 2,6,19,22,27,26,25,32,12,16\rbrace &{}\lbrace 2,1,21,22,27,26,31,32,12,11\rbrace &{}\lbrace 2,3,23,22,27,28,29,32,12,9\rbrace \\ &{}\lbrace 2,6,19,22,27,28,25,32,12,16\rbrace &{}\lbrace 2,1,21,22,27,30,31,32,12,11\rbrace &{}\lbrace 2,3,23,22,27,30,29,32,12,9\rbrace \\ &{}\lbrace 14,13,28,27,22,19,18,17,8,7\rbrace &{}\lbrace 14,15,26,27,22,19,20,17,8,5\rbrace &{}\lbrace 14,15,26,27,22,21,20,17,8,5\rbrace \\ &{}\lbrace 14,10,30,27,22,21,24,17,8,4\rbrace &{}\lbrace 14,10,30,27,22,23,24,17,8,4\rbrace &{}\lbrace 14,13,28,27,22,23,18,17,8,7\rbrace \\ &{}\lbrace 14,13,28,27,26,23,18,17,8,7\rbrace &{}\lbrace 14,10,30,27,26,23,24,17,8,4\rbrace &{}\lbrace 14,10,30,27,26,25,24,17,8,4\rbrace \\ &{}\lbrace 14,2,22,27,26,25,32,17,8,12\rbrace &{}\lbrace 14,2,22,27,26,31,32,17,8,12\rbrace &{}\lbrace 14,13,28,27,26,31,18,17,8,7\rbrace \\ &{}\lbrace 14,15,26,27,28,21,20,17,8,5\rbrace &{}\lbrace 14,10,30,27,28,21,24,17,8,4\rbrace &{}\lbrace 14,10,30,27,28,25,24,17,8,4\rbrace \\ &{}\lbrace 14,2,22,27,28,25,32,17,8,12\rbrace &{}\lbrace 14,2,22,27,28,29,32,17,8,12\rbrace &{}\lbrace 14,15,26,27,28,29,20,17,8,5\rbrace \\ &{}\lbrace 14,13,28,27,30,19,18,17,8,7\rbrace &{}\lbrace 14,15,26,27,30,19,20,17,8,5\rbrace &{}\lbrace 14,15,26,27,30,29,20,17,8,5\rbrace \\ &{}\lbrace 14,2,22,27,30,29,32,17,8,12\rbrace &{}\lbrace 14,13,28,27,30,31,18,17,8,7\rbrace &{}\lbrace 14,2,22,27,30,31,32,17,8,12\rbrace \\ &{}\lbrace 6,10,30,19,18,17,24,25,16,4\rbrace &{}\lbrace 6,2,22,19,18,17,32,25,16,12\rbrace &{}\lbrace 6,10,30,19,18,23,24,25,16,4\rbrace \\ &{}\lbrace 6,5,20,19,18,23,26,25,16,15\rbrace &{}\lbrace 6,5,20,19,18,31,26,25,16,15\rbrace &{}\lbrace 6,2,22,19,18,31,32,25,16,12\rbrace \\ &{}\lbrace 6,10,30,19,20,21,24,25,16,4\rbrace &{}\lbrace 6,7,18,19,20,21,28,25,16,13\rbrace &{}\lbrace 6,10,30,19,20,17,24,25,16,4\rbrace \\ \end{array} \end{aligned}$$
$$\begin{aligned} \begin{array}{llll} &{}\lbrace 6,2,22,19,20,17,32,25,16,12\rbrace &{}\lbrace 6,2,22,19,20,29,32,25,16,12\rbrace &{}\lbrace 6,7,18,19,20,29,28,25,16,13\rbrace \\ &{}\lbrace 6,10,30,19,22,21,24,25,16,4\rbrace &{}\lbrace 6,7,18,19,22,21,28,25,16,13\rbrace &{}\lbrace 6,10,30,19,22,23,24,25,16,4\rbrace \\ &{}\lbrace 6,5,20,19,22,23,26,25,16,15\rbrace &{}\lbrace 6,5,20,19,22,27,26,25,16,15\rbrace &{}\lbrace 6,7,18,19,22,27,28,25,16,13\rbrace \\ &{}\lbrace 6,5,20,19,30,27,26,25,16,15\rbrace &{}\lbrace 6,7,18,19,30,27,28,25,16,13\rbrace &{}\lbrace 6,7,18,19,30,29,28,25,16,13\rbrace \\ &{}\lbrace 6,2,22,19,30,29,32,25,16,12\rbrace &{}\lbrace 6,5,20,19,30,31,26,25,16,15\rbrace &{}\lbrace 6,2,22,19,30,31,32,25,16,12\rbrace \\ &{}\lbrace 13,14,27,28,21,20,17,18,7,8\rbrace &{}\lbrace 13,16,25,28,21,20,19,18,7,6\rbrace &{}\lbrace 13,16,25,28,21,22,19,18,7,6\rbrace \\ &{}\lbrace 13,9,29,28,21,22,23,18,7,3\rbrace &{}\lbrace 13,9,29,28,21,24,23,18,7,3\rbrace &{}\lbrace 13,14,27,28,21,24,17,18,7,8 \rbrace \\ &{}\lbrace 13,14,27,28,25,24,17,18,7,8\rbrace &{}\lbrace 13,9,29,28,25,24,23,18,7,3\rbrace &{}\lbrace 13,9,29,28,25,26,23,18,7,3 \rbrace \\ &{}\lbrace 13,1,21,28,25,26,31,18,7,11\rbrace &{}\lbrace 13,1,21,28,25,32,31,18,7,11\rbrace &{}\lbrace 13,14,27,28,25,32,17,18,7,8\rbrace \\ &{}\lbrace 13,16,25,28,27,22,19,18,7,6\rbrace &{}\lbrace 13,9,29,28,27,22,23,18,7,3\rbrace &{}\lbrace 13,9,29,28,27,26,23,18,7,3 \rbrace \\ &{}\lbrace 8,7,6,5,20,21,28,27,26,25\rbrace &{}\lbrace 8,7,6,5,20,29,28,27,26,25\rbrace &{}\lbrace 8,7,6,5,9,29,28,27,26,25\rbrace \\ &{}\lbrace 8,7,6,5,9,13,28,27,26,25\rbrace &{}\lbrace 8,7,6,5,1,13,28,27,26,25\rbrace &{}\lbrace 8,7,6,5,1,21,28,27,26,25\rbrace \\ &{}\lbrace 8,7,6,5,20,21,28,25,26,27\rbrace &{}\lbrace 8,7,6,5,20,29,28,25,26,27\rbrace &{}\lbrace 8,7,6,5,9,29,28,25,26,27\rbrace \\ &{}\lbrace 8,7,6,5,9,13,28,25,26,27\rbrace &{}\lbrace 8,7,6,5,1,13,28,25,26,27\rbrace &{}\lbrace 8,7,6,5,1,21,28,25,26,27\rbrace \\ &{}\lbrace 6,7,8,5,20,21,28,27,26,25\rbrace &{}\lbrace 6,7,8,5,20,29,28,27,26,25\rbrace &{}\lbrace 6,7,8,5,9,29,28,27,26,25\rbrace \\ &{}\lbrace 6,7,8,5,9,13,28,27,26,25\rbrace &{}\lbrace 6,7,8,5,1,13,28,27,26,25\rbrace &{}\lbrace 6,7,8,5,1,21,28,27,26,25\rbrace \\ &{}\lbrace 6,7,8,5,20,21,28,25,26,27\rbrace &{}\lbrace 6,7,8,5,20,29,28,25,26,27\rbrace &{}\lbrace 6,7,8,5,9,29,28,25,26,27\rbrace \\ &{}\lbrace 6,7,8,5,9,13,28,25,26,27\rbrace &{}\lbrace 6,7,8,5,1,13,28,25,26,27\rbrace &{}\lbrace 6,7,8,5,1,21,28,25,26,27\rbrace \\ &{}\lbrace 5,6,7,8,17,24,25,26,27,28\rbrace &{}\lbrace 5,6,7,8,17,32,25,26,27,28\rbrace &{}\lbrace 5,6,7,8,12,32,25,26,27,28\rbrace \\ &{}\lbrace 5,6,7,8,12,16,25,26,27,28\rbrace &{}\lbrace 5,6,7,8,4,16,25,26,27,28\rbrace &{}\lbrace 5,6,7,8,4,24,25,26,27,28\rbrace \\ &{}\lbrace 5,6,7,8,17,24,25,28,27,26\rbrace &{}\lbrace 5,6,7,8,17,32,25,28,27,26\rbrace &{}\lbrace 5,6,7,8,12,32,25,28,27,26\rbrace \\ &{}\lbrace 5,6,7,8,12,16,25,28,27,26\rbrace &{}\lbrace 5,6,7,8,4,16,25,28,27,26\rbrace &{}\lbrace 5,6,7,8,4,24,25,28,27,26\rbrace \\ &{}\lbrace 7,6,5,8,17,24,25,26,27,28\rbrace &{}\lbrace 7,6,5,8,17,32,25,26,27,28\rbrace &{}\lbrace 7,6,5,8,12,32,25,26,27,28\rbrace \\ &{}\lbrace 7,6,5,8,12,16,25,26,27,28\rbrace &{}\lbrace 7,6,5,8,4,16,25,26,27,28\rbrace &{}\lbrace 7,6,5,8,4,24,25,26,27,28\rbrace \\ &{}\lbrace 7,6,5,8,17,24,25,28,27,26\rbrace &{}\lbrace 7,6,5,8,17,32,25,28,27,26\rbrace &{}\lbrace 7,6,5,8,12,32,25,28,27,26\rbrace \\ &{}\lbrace 7,6,5,8,12,16,25,28,27,26\rbrace &{}\lbrace 7,6,5,8,4,16,25,28,27,26\rbrace &{}\lbrace 7,6,5,8,4,24,25,28,27,26\rbrace \\ &{}\lbrace 8,5,6,7,18,31,26,25,28,27\rbrace &{}\lbrace 8,5,6,7,18,23,26,25,28,27\rbrace &{}\lbrace 8,5,6,7,3,23,26,25,28,27\rbrace \\ &{}\lbrace 8,5,6,7,3,15,26,25,28,27\rbrace &{}\lbrace 8,5,6,7,11,31,26,25,28,27\rbrace &{}\lbrace 8,5,6,7,11,15,26,25,28,27\rbrace \\ &{}\lbrace 8,5,6,7,18,31,26,27,28,25\rbrace &{}\lbrace 8,5,6,7,18,23,26,27,28,25\rbrace &{}\lbrace 8,5,6,7,3,23,26,27,28,25\rbrace \\ &{}\lbrace 8,5,6,7,3,15,26,27,28,25\rbrace &{}\lbrace 8,5,6,7,11,31,26,27,28,25\rbrace &{}\lbrace 8,5,6,7,11,15,26,27,28,25\rbrace \\ &{}\lbrace 6,5,8,7,18,31,26,25,28,27\rbrace &{}\lbrace 6,5,8,7,18,23,26,25,28,27\rbrace &{}\lbrace 6,5,8,7,3,23,26,25,28,27\rbrace \\ &{}\lbrace 6,5,8,7,3,15,26,25,28,27\rbrace &{}\lbrace 6,5,8,7,11,31,26,25,28,27\rbrace &{}\lbrace 6,5,8,7,11,15,26,25,28,27\rbrace \\ &{}\lbrace 6,5,8,7,18,31,26,27,28,25\rbrace &{}\lbrace 6,5,8,7,18,23,26,27,28,25\rbrace &{}\lbrace 6,5,8,7,3,23,26,27,28,25\rbrace \\ &{}\lbrace 6,5,8,7,3,15,26,27,28,25\rbrace &{}\lbrace 6,5,8,7,11,31,26,27,28,25\rbrace &{}\lbrace 6,5,8,7,11,15,26,27,28,25\rbrace \\ &{}\lbrace 5,8,7,6,19,30,27,28,25,26\rbrace &{}\lbrace 5,8,7,6,19,22,27,28,25,26\rbrace &{}\lbrace 5,8,7,6,10,22,27,28,25,26\rbrace \\ &{}\lbrace 5,8,7,6,10,14,27,28,25,26\rbrace &{}\lbrace 5,8,7,6,2,30,27,28,25,26\rbrace &{}\lbrace 5,8,7,6,2,14,27,28,25,26\rbrace \\ &{}\lbrace 5,8,7,6,19,30,27,26,25,28\rbrace &{}\lbrace 5,8,7,6,19,22,27,26,25,28\rbrace &{}\lbrace 5,8,7,6,10,22,27,26,25,28\rbrace \\ &{}\lbrace 5,8,7,6,10,14,27,26,25,28\rbrace &{}\lbrace 5,8,7,6,2,30,27,26,25,28\rbrace &{}\lbrace 5,8,7,6,2,14,27,26,25,28\rbrace \\ &{}\lbrace 7,8,5,6,19,30,27,28,25,26\rbrace &{}\lbrace 7,8,5,6,19,22,27,28,25,26\rbrace &{}\lbrace 7,8,5,6,10,22,27,28,25,26\rbrace \\ &{}\lbrace 7,8,5,6,10,14,27,28,25,26\rbrace &{}\lbrace 7,8,5,6,2,14,27,28,25,26\rbrace &{}\lbrace 7,8,5,6,2,30,27,28,25,26\rbrace \\ &{}\lbrace 7,8,5,6,19,30,27,26,25,28\rbrace &{}\lbrace 7,8,5,6,19,22,27,26,25,28\rbrace &{}\lbrace 7,8,5,6,10,22,27,26,25,28\rbrace \\ &{}\lbrace 7,8,5,6,10,14,27,26,25,28\rbrace &{}\lbrace 7,8,5,6,2,14,27,26,25,28\rbrace &{}\lbrace 7,8,5,6,2,30,27,26,25,28\rbrace \\ &{}\lbrace 9,10,11,12,16,25,24,23,22,21\rbrace &{}\lbrace 9,10,11,12,16,4,24,23,22,21\rbrace &{}\lbrace 9,10,11,12,8,17,24,23,22,21\rbrace \\ &{}\lbrace 9,10,11,12,8,4,24,23,22,21\rbrace &{}\lbrace 9,10,11,12,32,17,24,23,22,21\rbrace &{}\lbrace 9,10,11,12,32,25,24,23,22,21\rbrace \\ &{}\lbrace 9,10,11,12,16,25,24,21,22,23\rbrace &{}\lbrace 9,10,11,12,16,4,24,21,22,23\rbrace &{}\lbrace 9,10,11,12,8,4,24,21,22,23\rbrace \\ &{}\lbrace 9,10,11,12,8,17,24,21,22,23\rbrace &{}\lbrace 9,10,11,12,32,17,24,21,22,23\rbrace &{}\lbrace 9,10,11,12,32,25,24,21,22,23\rbrace \\ &{}\lbrace 11,10,9,12,16,25,24,23,22,21\rbrace &{}\lbrace 11,10,9,12,16,4,24,23,22,21\rbrace &{}\lbrace 11,10,9,12,8,4,24,23,22,21\rbrace \\ &{}\lbrace 11,10,9,12,8,17,24,23,22,21\rbrace &{}\lbrace 11,10,9,12,32,17,24,23,22,21\rbrace &{}\lbrace 11,10,9,12,32,25,24,23,22,21\rbrace \\ &{}\lbrace 11,10,9,12,16,25,24,21,22,23\rbrace &{}\lbrace 11,10,9,12,16,4,24,21,22,23\rbrace &{}\lbrace 11,10,9,12,8,4,24,21,22,23\rbrace \\ &{}\lbrace 11,10,9,12,8,17,24,21,22,23\rbrace &{}\lbrace 11,10,9,12,32,17,24,21,22,23\rbrace &{}\lbrace 11,10,9,12,32,25,24,21,22,23\rbrace \\ &{}\lbrace 12,11,10,9,29,20,21,22,23,24\rbrace &{}\lbrace 12,11,10,9,29,28,,21,22,23,24\rbrace &{}\lbrace 12,11,10,9,13,28,21,22,23,24\rbrace \\ &{}\lbrace 12,11,10,9,13,1,21,22,23,24\rbrace &{}\lbrace 12,11,10,9,5,1,21,22,23,24\rbrace &{}\lbrace 12,11,10,9,5,20,21,22,23,24\rbrace \\ &{}\lbrace 12,11,10,9,29,20,21,24,23,22\rbrace &{}\lbrace 12,11,10,9,29,28,21,24,23,22\rbrace &{}\lbrace 12,11,10,9,13,28,21,24,23,22\rbrace \\ \end{array} \end{aligned}$$
$$\begin{aligned} \begin{array}{llll} &{}\lbrace 12,11,10,9,13,1,21,24,23,22\rbrace &{}\lbrace 12,11,10,9,5,1,21,24,23,22\rbrace &{}\lbrace 12,11,10,9,5,20,21,24,23,22\rbrace \\ &{}\lbrace 10,11,12,9,29,20,21,22,23,24\rbrace &{}\lbrace 10,11,12,9,29,28,21,22,23,24\rbrace &{}\lbrace 10,11,12,9,13,28,21,22,23,24\rbrace \\ &{}\lbrace 10,11,12,9,13,1,21,22,23,24\rbrace &{}\lbrace 10,11,12,9,5,1,21,22,23,24\rbrace &{}\lbrace 10,11,12,9,5,20,21,22,23,24\rbrace \\ &{}\lbrace 10,11,12,9,29,20,21,24,23,22\rbrace &{}\lbrace 10,11,12,9,29,28,21,24,23,22\rbrace &{}\lbrace 10,11,12,9,13,28,21,24,23,22\rbrace \\ &{}\lbrace 10,11,12,9,13,1,21,24,23,22\rbrace &{}\lbrace 10,11,12,9,5,1,21,24,23,22\rbrace &{}\lbrace 10,11,12,9,5,20,21,24,23,22\rbrace \\ &{}\lbrace 12,9,10,11,31,26,23,24,21,22\rbrace &{}\lbrace 12,9,10,11,31,18,23,24,21,22\rbrace &{}\lbrace 12,9,10,11,15,26,23,24,21,22\rbrace \\ &{}\lbrace 12,9,10,11,15,3,23,24,21,22\rbrace &{}\lbrace 12,9,10,11,7,3,23,24,21,22\rbrace &{}\lbrace 12,9,10,11,7,18,23,24,21,22\rbrace \\ &{}\lbrace 12,9,10,11,31,18,23,22,21,24\rbrace &{}\lbrace 12,9,10,11,31,26,23,22,21,24\rbrace &{}\lbrace 12,9,10,11,15,26,23,22,21,24\rbrace \\ &{}\lbrace 12,9,10,11,15,3,23,22,21,24\rbrace &{}\lbrace 12,9,10,11,7,3,23,22,21,24\rbrace &{}\lbrace 12,9,10,11,7,18,23,22,21,24\rbrace \\ &{}\lbrace 10,9,12,11,31,18,23,24,21,22\rbrace &{}\lbrace 10,9,12,11,31,26,23,24,21,22\rbrace &{}\lbrace 10,9,12,11,15,26,23,24,21,22\rbrace \\ &{}\lbrace 10,9,12,11,15,3,23,24,21,22\rbrace &{}\lbrace 10,9,12,11,7,3,23,24,21,22\rbrace &{}\lbrace 10,9,12,11,7,18,23,24,21,22\rbrace \\ &{}\lbrace 10,9,12,11,31,18,23,22,21,24\rbrace &{}\lbrace 10,9,12,11,31,26,23,22,21,24\rbrace &{}\lbrace 10,9,12,11,15,26,23,22,21,24\rbrace \\ &{}\lbrace 10,9,12,11,15,3,23,22,21,24\rbrace &{}\lbrace 10,9,12,11,7,3,23,22,21,24\rbrace &{}\lbrace 10,9,12,11,7,18,23,22,21,24\rbrace \\ &{}\lbrace 9,12,11,10,30,19,22,23,24,21\rbrace &{}\lbrace 9,12,11,10,30,27,22,23,24,21\rbrace &{}\lbrace 9,12,11,10,6,19,22,23,24,21\rbrace \\ &{}\lbrace 9,12,11,10,6,2,22,23,24,21\rbrace &{}\lbrace 9,12,11,10,14,27,22,23,24,21\rbrace &{}\lbrace 9,12,11,10,14,2,22,23,24,21\rbrace \\ &{}\lbrace 9,12,11,10,30,19,22,21,24,23\rbrace &{}\lbrace 9,12,11,10,30,27,22,21,24,23\rbrace &{}\lbrace 9,12,11,10,6,19,22,21,24,23\rbrace \\ &{}\lbrace 9,12,11,10,6,2,22,21,24,23\rbrace &{}\lbrace 9,12,11,10,14,2,22,21,24,23\rbrace &{}\lbrace 9,12,11,10,14,27,22,21,24,23\rbrace \\ &{}\lbrace 11,12,9,10,30,19,22,23,24,21\rbrace &{}\lbrace 11,12,9,10,30,27,22,23,24,21\rbrace &{}\lbrace 11,12,9,10,6,19,22,23,24,21\rbrace \\ &{}\lbrace 11,12,9,10,6,2,22,23,24,21\rbrace &{}\lbrace 11,12,9,10,14,2,22,23,24,21\rbrace &{}\lbrace 11,12,9,10,14,27,22,23,24,21\rbrace \\ &{}\lbrace 11,12,9,10,30,19,22,21,24,23\rbrace &{}\lbrace 11,12,9,10,30,27,22,21,24,23\rbrace &{}\lbrace 11,12,9,10,6,19,22,21,24,23\rbrace \\ &{}\lbrace 11,12,9,10,6,2,22,21,24,23\rbrace &{}\lbrace 11,12,9,10,14,2,22,21,24,23\rbrace &{}\lbrace 11,12,9,10,14,27,22,21,24,23\rbrace \\ &{}\lbrace 4,3,2,1,21,20,29,30,31,32\rbrace &{}\lbrace 4,3,2,1,21,28,29,30,31,32\rbrace &{}\lbrace 4,3,2,1,5,20,29,30,31,32\rbrace \\ &{}\lbrace 4,3,2,1,5,9,29,30,31,32\rbrace &{}\lbrace 4,3,2,1,13,9,29,30,31,32\rbrace &{}\lbrace 4,3,2,1,13,28,29,30,31,32\rbrace \\ &{}\lbrace 4,3,2,1,21,20,29,32,31,30\rbrace &{}\lbrace 4,3,2,1,21,28,29,32,31,30\rbrace &{}\lbrace 4,3,2,1,5,20,29,32,31,30\rbrace \\ &{}\lbrace 4,3,2,1,5,9,29,32,31,30\rbrace &{}\lbrace 4,3,2,1,13,9,29,32,31,30\rbrace &{}\lbrace 4,3,2,1,13,28,29,32,31,30\rbrace \\ &{}\lbrace 2,3,4,1,21,20,29,30,31,32\rbrace &{}\lbrace 2,3,4,1,21,28,29,30,31,32\rbrace &{}\lbrace 2,3,4,1,5,20,29,30,31,32\rbrace \\ &{}\lbrace 2,3,4,1,5,9,29,30,31,32\rbrace &{}\lbrace 2,3,4,1,13,9,29,30,31,32\rbrace &{}\lbrace 2,3,4,1,13,28,29,30,31,32\rbrace \\ &{}\lbrace 2,3,4,1,21,20,29,32,31,30\rbrace &{}\lbrace 2,3,4,1,21,28,29,32,31,30\rbrace &{}\lbrace 2,3,4,1,5,20,29,32,31,30\rbrace \\ &{}\lbrace 2,3,4,1,5,9,29,32,31,30\rbrace &{}\lbrace 2,3,4,1,13,9,29,32,31,30\rbrace &{}\lbrace 2,3,4,1,13,28,29,32,31,30\rbrace \\ &{}\lbrace 1,2,3,4,24,17,32,31,30,29\rbrace &{}\lbrace 1,2,3,4,24,25,32,31,30,29\rbrace &{}\lbrace 1,2,3,4,16,25,32,31,30,29\rbrace \\ &{}\lbrace 1,2,3,4,16,12,32,31,30,29\rbrace &{}\lbrace 1,2,3,4,8,12,32,31,30,29\rbrace &{}\lbrace 1,2,3,4,8,17,32,31,30,29\rbrace \\ &{}\lbrace 1,2,3,4,24,17,32,29,30,31\rbrace &{}\lbrace 1,2,3,4,24,25,32,29,30,31\rbrace &{}\lbrace 1,2,3,4,16,25,32,29,30,31\rbrace \\ &{}\lbrace 1,2,3,4,16,12,32,29,30,31\rbrace &{}\lbrace 1,2,3,4,8,12,32,29,30,31\rbrace &{}\lbrace 1,2,3,4,8,17,32,29,30,31\rbrace \\ &{}\lbrace 3,2,1,4,24,17,32,31,30,29\rbrace &{}\lbrace 3,2,1,4,24,25,32,31,30,29\rbrace &{}\lbrace 3,2,1,4,16,25,32,31,30,29\rbrace \\ &{}\lbrace 3,2,1,4,16,12,32,31,30,29\rbrace &{}\lbrace 3,2,1,4,8,12,32,31,30,29\rbrace &{}\lbrace 3,2,1,4,8,17,32,31,30,29\rbrace \\ &{}\lbrace 3,2,1,4,24,17,32,29,30,31\rbrace &{}\lbrace 3,2,1,4,24,25,32,29,30,31\rbrace &{}\lbrace 3,2,1,4,16,25,32,29,30,31\rbrace \\ &{}\lbrace 3,2,1,4,16,12,32,29,30,31\rbrace &{}\lbrace 3,2,1,4,8,12,32,29,30,31\rbrace &{}\lbrace 3,2,1,4,8,17,32,29,30,31\rbrace \\ &{}\lbrace 1,4,3,2,22,27,30,31,32,29\rbrace &{}\lbrace 1,4,3,2,22,19,30,31,32,29\rbrace &{}\lbrace 1,4,3,2,6,19,30,31,32,29\rbrace \\ &{}\lbrace 1,4,3,2,6,10,30,31,32,29\rbrace &{}\lbrace 1,4,3,2,14,10,30,31,32,29\rbrace &{}\lbrace 1,4,3,2,14,27,30,31,32,29\rbrace \\ &{}\lbrace 1,4,3,2,22,27,30,29,32,31\rbrace &{}\lbrace 1,4,3,2,22,19,30,29,32,31\rbrace &{}\lbrace 1,4,3,2,6,19,30,29,32,31\rbrace \\ &{}\lbrace 1,4,3,2,6,10,30,29,32,31\rbrace &{}\lbrace 1,4,3,2,14,10,30,29,32,31\rbrace &{}\lbrace 1,4,3,2,14,27,30,29,32,31\rbrace \\ &{}\lbrace 3,4,1,2,22,27,30,31,32,29\rbrace &{}\lbrace 3,4,1,2,22,19,30,31,32,29\rbrace &{}\lbrace 3,4,1,2,6,19,30,31,32,29\rbrace \\ &{}\lbrace 3,4,1,2,6,10,30,31,32,29\rbrace &{}\lbrace 3,4,1,2,14,10,30,31,32,29\rbrace &{}\lbrace 3,4,1,2,14,27,30,31,32,29\rbrace \\ &{}\lbrace 3,4,1,2,22,27,30,29,32,31\rbrace &{}\lbrace 3,4,1,2,22,19,30,29,32,31\rbrace &{}\lbrace 3,4,1,2,6,19,30,29,32,31\rbrace \\ &{}\lbrace 3,4,1,2,6,10,30,29,32,31\rbrace &{}\lbrace 3,4,1,2,14,10,30,29,32,31\rbrace &{}\lbrace 3,4,1,2,14,27,30,29,32,31\rbrace \\ &{}\lbrace 2,1,4,3,23,26,31,32,29,30\rbrace &{}\lbrace 2,1,4,3,23,18,31,32,29,30\rbrace &{}\lbrace 2,1,4,3,15,26,31,32,29,30\rbrace \\ &{}\lbrace 2,1,4,3,15,11,31,32,29,30\rbrace &{}\lbrace 2,1,4,3,7,18,31,32,29,30\rbrace &{}\lbrace 2,1,4,3,7,11,31,32,29,30\rbrace \\ &{}\lbrace 2,1,4,3,23,26,31,30,29,32\rbrace &{}\lbrace 2,1,4,3,23,18,31,30,29,32\rbrace &{}\lbrace 2,1,4,3,15,26,31,30,29,32\rbrace \\ &{}\lbrace 2,1,4,3,15,11,31,30,29,32\rbrace &{}\lbrace 2,1,4,3,7,18,31,30,29,32\rbrace &{}\lbrace 2,1,4,3,7,11,31,30,29,32\rbrace \\ &{}\lbrace 4,1,2,3,23,26,31,32,29,30\rbrace &{}\lbrace 4,1,2,3,23,18,31,32,29,30\rbrace &{}\lbrace 4,1,2,3,15,26,31,32,29,30\rbrace \\ &{}\lbrace 4,1,2,3,15,11,31,32,29,30\rbrace &{}\lbrace 4,1,2,3,7,18,31,32,29,30\rbrace &{}\lbrace 4,1,2,3,7,11,31,32,29,30\rbrace \\ &{}\lbrace 4,1,2,3,23,26,31,30,29,32\rbrace &{}\lbrace 4,1,2,3,23,18,31,30,29,32\rbrace &{}\lbrace 4,1,2,3,15,26,31,30,29,32\rbrace \\ &{}\lbrace 4,1,2,3,15,11,31,30,29,32\rbrace &{}\lbrace 4,1,2,3,7,11,31,30,29,32\rbrace &{}\lbrace 4,1,2,3,7,18,31,30,29,32\rbrace \\ &{}\lbrace 13,14,15,16,12,32,17,18,19,20\rbrace &{}\lbrace 13,14,15,16,12,8,17,18,19,20\rbrace &{}\lbrace 13,14,15,16,4,24,17,18,19,20\rbrace \\ &{}\lbrace 13,14,15,16,4,8,17,18,19,20\rbrace &{}\lbrace 13,14,15,16,25,24,17,18,19,20\rbrace &{}\lbrace 13,14,15,16,25,32,17,18,19,20\rbrace \\ \end{array} \end{aligned}$$
$$\begin{aligned} \begin{array}{llll} &{}\lbrace 13,14,15,16,12,32,17,20,19,18\rbrace &{}\lbrace 13,14,15,16,12,8,17,20,19,18\rbrace &{}\lbrace 13,14,15,16,4,8,17,20,19,18\rbrace \\ &{}\lbrace 13,14,15,16,4,24,17,20,19,18\rbrace &{}\lbrace 13,14,15,16,25,24,17,20,19,18\rbrace &{}\lbrace 13,14,15,16,25,32,17,20,19,18\rbrace \\ &{}\lbrace 15,14,13,16,12,32,17,18,19,20\rbrace &{}\lbrace 15,14,13,16,12,8,17,18,19,20\rbrace &{}\lbrace 15,14,13,16,4,8,17,18,19,20\rbrace \\ &{}\lbrace 15,14,13,16,4,24,17,18,19,20\rbrace &{}\lbrace 15,14,13,16,25,24,17,18,19,20\rbrace &{}\lbrace 15,14,13,16,25,32,17,18,19,20\rbrace \\ &{}\lbrace 15,14,13,16,12,32,17,20,19,18\rbrace &{}\lbrace 15,14,13,16,12,8,17,20,19,18\rbrace &{}\lbrace 15,14,13,16,4,8,17,20,19,18\rbrace \\ &{}\lbrace 15,14,13,16,4,24,17,20,19,18\rbrace &{}\lbrace 15,14,13,16,25,24,17,20,19,18\rbrace &{}\lbrace 15,14,13,16,25,32,17,20,19,18\rbrace \\ &{}\lbrace 16,15,14,13,28,29,20,19,18,17\rbrace &{}\lbrace 16,15,14,13,28,21,20,19,18,17\rbrace &{}\lbrace 16,15,14,13,9,29,20,19,18,17\rbrace \\ &{}\lbrace 16,15,14,13,9,5,20,19,18,17\rbrace &{}\lbrace 16,15,14,13,1,5,20,19,18,17\rbrace &{}\lbrace 16,15,14,13,1,21,20,19,18,17\rbrace \\ &{}\lbrace 16,15,14,13,28,29,20,17,18,19\rbrace &{}\lbrace 16,15,14,13,28,21,20,17,18,19\rbrace &{}\lbrace 16,15,14,13,9,29,20,17,18,19\rbrace \\ &{}\lbrace 16,15,14,13,9,5,20,17,18,19\rbrace &{}\lbrace 16,15,14,13,1,5,20,17,18,19\rbrace &{}\lbrace 16,15,14,13,1,21,20,17,18,19\rbrace \\ &{}\lbrace 14,15,16,13,28,29,20,19,18,17\rbrace &{}\lbrace 14,15,16,13,28,21,20,19,18,17\rbrace &{}\lbrace 14,15,16,13,9,29,20,19,18,17\rbrace \\ &{}\lbrace 14,15,16,13,9,5,20,19,18,17\rbrace &{}\lbrace 14,15,16,13,1,5,20,19,18,17\rbrace &{}\lbrace 14,15,16,13,1,21,20,19,18,17\rbrace \\ &{}\lbrace 14,15,16,13,28,29,20,17,18,19\rbrace &{}\lbrace 14,15,16,13,28,21,20,17,18,19\rbrace &{}\lbrace 14,15,16,13,9,29,20,17,18,19\rbrace \\ &{}\lbrace 14,15,16,13,9,5,20,17,18,19\rbrace &{}\lbrace 14,15,16,13,1,5,20,17,18,19\rbrace &{}\lbrace 14,15,16,13,1,21,20,17,18,19\rbrace \\ &{}\lbrace 15,16,13,14,27,22,19,18,17,20\rbrace &{}\lbrace 15,16,13,14,27,30,19,18,17,20\rbrace &{}\lbrace 15,16,13,14,2,22,19,18,17,20\rbrace \\ &{}\lbrace 15,16,13,14,2,6,19,18,17,20\rbrace &{}\lbrace 15,16,13,14,10,30,19,18,17,20\rbrace &{}\lbrace 15,16,13,14,10,6,19,18,17,20\rbrace \\ &{}\lbrace 15,16,13,14,27,22,19,20,17,18\rbrace &{}\lbrace 15,16,13,14,27,30,19,20,17,18\rbrace &{}\lbrace 15,16,13,14,2,22,19,20,17,18\rbrace \\ &{}\lbrace 15,16,13,14,2,6,19,20,17,18\rbrace &{}\lbrace 15,16,13,14,10,6,19,20,17,18\rbrace &{}\lbrace 15,16,13,14,10,30,19,20,17,18\rbrace \\ &{}\lbrace 13,16,15,14,27,22,19,18,17,20\rbrace &{}\lbrace 13,16,15,14,27,30,19,18,17,20\rbrace &{}\lbrace 13,16,15,14,2,22,19,18,17,20\rbrace \\ &{}\lbrace 13,16,15,14,2,6,19,18,17,20\rbrace &{}\lbrace 13,16,15,14,10,6,19,18,17,20\rbrace &{}\lbrace 13,16,15,14,10,30,19,18,17,20\rbrace \\ &{}\lbrace 13,16,15,14,27,22,19,20,17,18\rbrace &{}\lbrace 13,16,15,14,27,30,19,20,17,18\rbrace &{}\lbrace 13,16,15,14,2,22,19,20,17,18\rbrace \\ &{}\lbrace 13,16,15,14,2,6,19,20,17,18\rbrace &{}\lbrace 13,16,15,14,10,6,19,20,17,18\rbrace &{}\lbrace 13,16,15,14,10,30,19,20,17,18\rbrace \\ &{}\lbrace 16,13,14,15,26,23,18,19,20,17\rbrace &{}\lbrace 16,13,14,15,26,31,18,19,20,17\rbrace &{}\lbrace 16,13,14,15,3,23,18,19,20,17\rbrace \\ &{}\lbrace 16,13,14,15,3,7,18,19,20,17\rbrace &{}\lbrace 16,13,14,15,11,7,18,19,20,17\rbrace &{}\lbrace 16,13,14,15,11,31,18,19,20,17\rbrace \\ &{}\lbrace 16,13,14,15,26,23,18,17,20,19\rbrace &{}\lbrace 16,13,14,15,26,31,18,17,20,19\rbrace &{}\lbrace 16,13,14,15,3,23,18,17,20,19\rbrace \\ &{}\lbrace 16,13,14,15,3,7,18,17,20,19\rbrace &{}\lbrace 16,13,14,15,11,7,18,17,20,19\rbrace &{}\lbrace 16,13,14,15,11,31,18,17,20,19\rbrace \\ &{}\lbrace 14,13,16,15,26,23,18,19,20,17\rbrace &{}\lbrace 14,13,16,15,26,31,18,19,20,17\rbrace &{}\lbrace 14,13,16,15,3,23,18,19,20,17\rbrace \\ &{}\lbrace 14,13,16,15,3,7,18,19,20,17\rbrace &{}\lbrace 14,13,16,15,11,7,18,19,20,17\rbrace &{}\lbrace 14,13,16,15,11,31,18,19,20,17\rbrace \\ &{}\lbrace 14,13,16,15,26,23,18,17,20,19\rbrace &{}\lbrace 14,13,16,15,26,31,18,17,20,19\rbrace &{}\lbrace 14,13,16,15,3,23,18,17,20,19\rbrace \\ &{}\lbrace 14,13,16,15,3,7,18,17,20,19\rbrace &{}\lbrace 14,13,16,15,11,7,18,17,20,19\rbrace &{}\lbrace 14,13,16,15,11,31,18,17,20,19\rbrace \end{array} \end{aligned}$$

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Huilgol, M.I., Vidya, M.D. PBIB-designs and association schemes arising from minimum bi-connected dominating sets of hypercubes. Afr. Mat. 31, 1009–1028 (2020). https://doi.org/10.1007/s13370-020-00777-2

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