Proceedings of the Forty-Fourth Annual ACM Symposium on Theory of Computing 2012
DOI: 10.1145/2213977.2214075
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Probabilistic existence of rigid combinatorial structures

Abstract: We show the existence of rigid combinatorial objects which previously were not known to exist. Specifically, for a wide range of the underlying parameters, we show the existence of non-trivial orthogonal arrays, t-designs, and t-wise permutations. In all cases, the sizes of the objects are optimal up to polynomial overhead. The proof of existence is probabilistic. We show that a randomly chosen such object has the required properties with positive yet tiny probability. The main technical ingredient is a specia… Show more

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Cited by 16 publications
(32 citation statements)
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“…For small t and large n, this result is close to the simple lower bound of n(n − 1) · · · (n − t + 1) which is implied by (1). It is important to emphasize, however, that the proof in [7] is purely existential and provides no hint as to the construction of such small t-wise uniform sets. Our work gives the first non-trivial explicit construction of an infinite family of t-wise uniform sets for t ≥ 4.…”
Section: Introductionsupporting
confidence: 54%
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“…For small t and large n, this result is close to the simple lower bound of n(n − 1) · · · (n − t + 1) which is implied by (1). It is important to emphasize, however, that the proof in [7] is purely existential and provides no hint as to the construction of such small t-wise uniform sets. Our work gives the first non-trivial explicit construction of an infinite family of t-wise uniform sets for t ≥ 4.…”
Section: Introductionsupporting
confidence: 54%
“…Moreover, it is known (see for example [3,Theorem 5.2]) that for n ≥ 25 and t ≥ 4 there are no subgroups of S n , other than A n and S n itself, that form a t-wise uniform set; such subgroups are called t-transitive subgroups of S n . In contrast, it was shown recently [7] that for all n ≥ 1 and 1 ≤ t ≤ n, there exists a t-wise uniform set of permutations on n letters of size n ct for some universal constant c > 0. For small t and large n, this result is close to the simple lower bound of n(n − 1) · · · (n − t + 1) which is implied by (1).…”
Section: Introductionmentioning
confidence: 96%
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“…This may be super-exponential in the input size. Like for the equality protocol, we can optimize by taking advantage of k-wise independent families of permutations, as these may have smaller descriptions (the existence of small permutation families with this property was recently proven in [25], but this is only an existential result. Instead we can use the efficient explicit constructions of [23] but achieve only statistical security).…”
Section: Theorem 7 the Protocol Inmentioning
confidence: 99%