Linear Time Probabilistic Erasure Codes With Nearly Optimal Recovery

We present a linear time probabilistic erasure code with the following properties. The encoding algorithm produces cn letters from an n letter message. The decoding algorithm is able with high probability to reconstruct the message from any set of (1+ ǫ)n letters. Alon and Luby [5, 6] simultaneously developed a deterministic version but their running time is O(ǫ−4n) instead of our O(ǫ−1 ln(ǫ)n) time. We also decrease the minimum packet size from many to one letter. In some sense this erasure code technology has been completely subsumed by the latest generation of RaptorQ codes [12, 10]. On the other hand there are still some good ideas in this paper. One such idea is matching upper and lower bounds on the cost of a particular game having to do with randomly throwing balls into a hierarchy of bins. This game has application to information theory.