TY - GEN
T1 - Probabilistic existence of rigid combinatorial structures
AU - Kuperberg, Greg
AU - Lovett, Shachar
AU - Peled, Ron
PY - 2012
Y1 - 2012
N2 - 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 special local central limit theorem for suitable lattice random walks with finitely many steps.
AB - 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 special local central limit theorem for suitable lattice random walks with finitely many steps.
KW - designs
KW - local central limit theorem
KW - orthogonal arrays
KW - permutations
KW - probabilistic method
UR - http://www.scopus.com/inward/record.url?scp=84862632182&partnerID=8YFLogxK
U2 - 10.1145/2213977.2214075
DO - 10.1145/2213977.2214075
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AN - SCOPUS:84862632182
SN - 9781450312455
T3 - Proceedings of the Annual ACM Symposium on Theory of Computing
SP - 1091
EP - 1105
BT - STOC '12 - Proceedings of the 2012 ACM Symposium on Theory of Computing
T2 - 44th Annual ACM Symposium on Theory of Computing, STOC '12
Y2 - 19 May 2012 through 22 May 2012
ER -