Raptor Codes AMIN SHOKROLLAHI ∗ Laboratoire d’algorithmique Laboratoire des mathematiques algorithmiques ´ Ecole Polytechnique F´ ed´ erale de Lausanne 1015 Lausanne, Switzerland amin.shokrollahi@epfl.ch September 19, 2005 Abstract LT-Codes are a new class of codes introduced in [1] for the purpose of scalable and fault-tolerant distribution of data over computer networks. In this paper we introduce Raptor Codes, an extension of LT-Codes with linear time encoding and decoding. We will exhibit a class of universal Raptor codes: for a given integer k, and any real ε> 0, Raptor codes in this class produce a potentially infinite stream of symbols such that any subset of symbols of size k(1 + ε) is sufficient to recover the original k symbols with high probability. Each output symbol is generated using O(log(1/ε)) operations, and the original symbols are recovered from the collected ones with O(k log(1/ε)) operations. We will also introduce novel techniques for the analysis of the error probability of the decoder for finite length Raptor codes. Moreover, we will introduce and analyze systematic versions of Raptor codes, i.e., versions in which the first output elements of the coding system coincide with the original k elements. 1 Introduction The binary erasure channel (BEC) of communication was introduced by Elias [2] in 1955, but it was regarded as a rather theoretical channel model until the large-scale deployment of the Internet about 40 years later. ∗ Work on this project was done while the author was a full time employee of Digital Fountain, Inc. 1