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Exact Covering Configurations and Steiner Systems
Author(s) -
Hartman A.,
Mullin R. C.,
Stinson D. R.
Publication year - 1982
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-25.2.193
Subject(s) - combinatorics , cardinality (data modeling) , mathematics , prime (order theory) , prime power , plane (geometry) , discrete mathematics , integer (computer science) , computer science , geometry , programming language , data mining
An exact (λ, t )‐covering, B , of a v ‐set X , is a collection of proper subsets of X , with the property that each t ‐subset of X is contained in exactly λ members of B . Let g (λ, t, v ) denote the minimum cardinality | B | of an exact (λ, t )‐covering of X . We show that if q ⩾ 3 is a prime power then g (1, 3, q 2 + 1) = q 3 + q , the minimal configuration being an inversive plane. We also show that g (1, 3, q 2 − α) = q 3 + q for q ⩾ 4 and α 0, where α is small relative to q . A few isolated values of g (1, t, v ) for t > 3 are also obtained.