Superimposed codes are almost big distance ones

Let T(r,n) denote the maximum number of subsets of an n-set satisfying the condition that no set is covered by the union of r others, while let T*(/spl epsiv/,n) be the maximum size of a n/sub 0/(r)logT(r,n)/n/spl les/logT*(loglogr/r,n)/n.