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Large sets with multiplicity

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Abstract

Large sets of combinatorial designs has always been a fascinating topic in design theory. These designs form a partition of the whole space into combinatorial designs with the same parameters. In particular, a large set of block designs, whose blocks are of size k taken from an n-set, is a partition of all the k-subsets of the n-set into disjoint copies of block designs, defined on the n-set, and with the same parameters. The current most intriguing question in this direction is whether large sets of Steiner quadruple systems exist and to provide explicit constructions for those parameters for which they exist. In view of its difficulty no one ever presented an explicit construction even for one nontrivial order. Hence, we seek for related generalizations. As generalizations, to the existence question of large sets, we consider two related questions. The first one is to provide constructions for sets on Steiner systems in which each block (quadruple or a k-subset) is contained in exactly \(\mu \) systems. The constructions of such systems also yield secure protocols for the generalized Russian cards problem. The second question is to provide constructions for large set of H-designs (mainly for quadruples, but also for larger block size), which have applications in threshold schemes and in quantum jump codes. We prove the existence of such systems for many parameters using orthogonal arrays, perpendicular arrays, ordered designs, sets of permutations, and one-factorizations of the complete graph.

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Acknowledgements

We are indebted to Doug Stinson who brought to our attention his paper [45] on the Russian cards problem. This problem has given further motivation for our constructions of large sets with multiplicity. We are also indebted to a reviewer which read our draft very carefully and gave many constructive comments which amended this paper.

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Authors and Affiliations

  1. Department of Computer Science, Technion, Haifa, 3200003, Israel

    Tuvi Etzion

  2. Department of Mathematics, Beijing Jiaotong University, Beijing, China

    Junling Zhou

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  1. Tuvi Etzion
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Correspondence to Tuvi Etzion.

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Communicated by L. Teirlinck.

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This research was supported in part by the 111 Project of China (B16002) and in part by the NSFC grants 11571034 and 11971053. Part of the research was performed during a visit of T. Etzion to Beijing Jiaotong University. He expresses sincere thanks to the 111 Project of China (B16002) for its support and to the Department of Mathematics at Beijing Jiaotong University for their kind hospitality. T. Etzion was also supported in part by the Bernard Elkin Chair in Computer Science.

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Etzion, T., Zhou, J. Large sets with multiplicity. Des. Codes Cryptogr. 89, 1661–1690 (2021). https://doi.org/10.1007/s10623-021-00878-4

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