Helper data schemes are a security primitive used for privacy-preserving biometric databases and physical unclonable functions. One of the oldest known helper data schemes is the code offset method (COM). We propose an extension of the COM: the helper data are accompanied by many instances of fake helper data that are drawn from the same distribution as the real one. While the adversary has no way to distinguish between them, the legitimate party has more information and can see the difference. We use a low-density parity check code in order to improve the efficiency of the legitimate party's selection procedure. Our construction provides a new kind of tradeoff: more effective use of the source entropy, at the price of increased helper data storage. We give a security analysis in terms of Shannon entropy and order-2 Rényi entropy. We also propose a variant of our scheme in which the helper data list is not stored but pseudorandomly generated, changing the tradeoff to source entropy utilization versus computation effort.