Guess & check codes for deletions and synchronization

We consider the problem of constructing codes that can correct δ deletions occurring in an arbitrary binary string of length n bits. Varshamov-Tenengolts (VT) codes can correct all possible single deletions (δ = 1) with an asymptotically optimal redundancy. Finding similar codes for δ ≥ 2 deletions is an open problem. We propose a new family of codes, that we call Guess & Check (GC) codes, that can correct, with high probability, a constant number of deletions δ occurring at uniformly random positions within an arbitrary string. The GC codes are based on MDS codes and have an asymptotically optimal redundancy that is Θ(δ logn). We provide deterministic polynomial time encoding and decoding schemes for these codes. We also describe the applications of GC codes to file synchronization.