We consider binary fingerprinting codes that trace at least one of t pirates using the marking assumption. Ensembles of binary equal-weight codes are considered along with a new efficient decoding algorithm. The design substantially increases the code rates of the former fingerprinting constructions. In particular, for large t, the new tfingerprinting codes have code rate of t−2 ln 2 and identify a pirate with an error probability that declines exponentially in code length n.