Summary: Anti-Hadamard matrices, coin weighing, threshold gates and
Van H. V~u
February 22, 2002
Let 1(n) denote the maximum possible absolute value of an entry of the inverse of
an n by n invertible matrix with 0, 1 entries. It is proved that 1(n) = n( 1
solves a problem of Graham and Sloane.
Let m(n) denote the maximum possible number m such that given a set of m coins
out of a collection of coins of two unknown distinct weights, one can decide if all the
coins have the same weight or not using n weighings in a regular balance beam. It is
shown that m(n) = n( 1
. This settles a problem of Kozlov and V~u.
Let D(n) denote the maximum possible degree of a regular multi-hypergraph on n
vertices that contains no proper regular nonempty subhypergraph. It is shown that
D(n) = n( 1