Efficient Erasure Correcting Codes

Authors: Michael G. Luby, Michael Mitzenmacher, M. Amin Shokrollahi, and Daniel A. Spielman.

Bibliographic Information: Appeared in IEEE Transactions on Information Theory, 47(2), pp. 569-584, Feb. 2001. Preliminary version appeared in the The Twenty-Ninth Annual ACM Symposium on Theory of Computing (STOC) under the title "Practial Erasure-Resilient Codes" and with Volker Stemann as a co-author.


The codes presented in this paper have been dubbed Tornado Codes and have been examined for use in compensating for packet loss in internet traffic, especially in multicast video.


We present randomized constructions of linear-time encodable and decodable codes that can transmit over lossy channels at rates extremely close to capacity. The encoding and decoding algorithms for these codes have fast and simple software implementations. Partial implementations of our algorithms are faster by orders of magnitude than the best software implementations of any previous algorithm for this problem. We expect these codes will be extremely useful for applications such as real-time audio and video transmission over the Internet, where lossy channels are common and fast decoding is a requirement.

Despite the simplicity of the algorithms, their design and analysis are mathematically intricate. The design requires the careful choice of a random irregular bipartite graph, where the structure of the irregular graph is extremely important. We model the progress of the decoding algorithm by a set of differential equations. The solution to these equations can then be expressed as polynomials in one variable with coefficients determined by the graph structure. Based on these polynomials, we design a graph structure that guarantees successful decoding with high probability.

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Daniel A. Spielman
Last modified: Fri Aug 24 15:47:38 2001