Anti-Hadamard Matrices, Coin Weighing, Threshold Gates, and Indecomposable Hypergraphs

Letχ1(n) denote the maximum possible absolute value of an entry of the inverse of annbyninvertible matrix with 0,1 entries. It is proved thatχ1(n)=n(1/2+o(1)) n. This solves a problem of Graham and Sloane. Letm(n) denote the maximum possible numbermsuch that given a set ofmcoins out of a collection of coins of two unknown distinct weights, one can decide if all the coins have the same weight or not usingnweighings in a regular balance beam. It is shown thatm(n)=n(1/2+o(1)) n. This settles a problem of Kozlov and Vũ. LetD(n) denote the maximum possible degree of a regular multi-hypergraph onnvertices that contains no proper regular nonempty subhypergraph. It is shown thatD(n)=n(1/2+o(1)) n. This improves estimates of Shapley, van Lint and Pollak. All these results and several related ones are proved by a similar technique whose main ingredient is an extension of a construction of Håstad of threshold gates that require large weights.