Efficient erasure correcting codes

Abstract

Abstract-We introduce a simple erasure recovery algorithm for codes derived from cascades of sparse bipartite graphs and analyze the algorithm by analyzing a corresponding discrete-time random process. As a result, we obtain a simple criterion involving the fractions of nodes of different degrees on both sides of the graph which is necessary and sufficient for the decoding process to finish successfully with high probability. By carefully designing these graphs we can construct for any given rate and any given real number a family of linear codes of rate which can be encoded in time proportional to ln(1 ) times their block length . Furthermore, a codeword can be recovered with high probability from a portion of its entries of length (1 + ) or more. The recovery algorithm also runs in time proportional to ln(1 ). Our algorithms have been implemented and work well in practice; various implementation issues are discussed.

Dynamic phase transition for decoding algorithms

“…Amazingly good codes [37][38][39] where obtained by slightly more sophisticated irregular constructions. In this case one assigns the distributions of the degrees of parity check nodes and variable nodes in the Tanner graph.…”

Maximum matchings in random bipartite graphs and the space utilization of Cuckoo Hash tables

“…Andrea Montanari has pointed out that our proof of Theorem 2 via the differential equations method is not new and already appears in Luby, Mitzenmacher, Shokrollahi and Spielman [13] and also in Dembo and Montanari [8]. We will prune this from the final version of the paper, but leave it in here for now.…”

On the Security of LBlock against the Cube Attack and Side Channel Cube Attack

“…The process may be made possible via physical probing, power measurement, or any other type of side channel. However, this information is quite noisy and this problem is addressed by using error correction techniques like erasure codes [73].…”