  Booth and Cox proposed the E(s^2) criterion for constructing two-level
supersaturated designs. Nguyen [Technometrics 38 (1996) 69-73] and Tang and Wu
[Canad. J. Statist 25 (1997) 191-201] independently derived a lower bound for
E(s^2). This lower bound can be achieved only when m is a multiple of N-1,
where m is the number of factors and N is the run size. We present a method
that uses difference families to construct designs that satisfy this lower
bound. We also derive better lower bounds for the case where the Nguyen-Tang-Wu
bound is not achievable. Our bounds cover more cases than a bound recently
obtained by Butler, Mead, Eskridge and Gilmour [J.
  R. Stat. Soc. Ser. B Stat. Methodol. 63 (2001) 621-632]. New E(s^2)-optimal
designs are obtained by using a computer to search for designs that achieve the
improved bounds.
