Exact coverings of triples with specified longest block length

Abstract

AbstractA minimal (1,3; ν) covering occurs when we have a family of proper subsets selected from ν elements with the property that every triple occurs exactly once in the family and no family of smaller cardinality possesses this property. Woodall developed a lower bound W for the quantity g(k)(1, 3; ν) which represents the cardinality of a minimal family with longest block of length k. The Woodall bound is only accurate in the region when k ≥ ν/2. We develop an expression for the excess of the true value over the Woodall bound and apply this to show that, when k ≥ ν/2, the value of g(1,3; ν) = W + 1 when k is even and W + 1 + when k is odd.