Summary of Raptor Codes

A fountain code produces for given set of k input symbols (x1, . . . , xk) a potentially limitless stream of output symbols z1, z2, . . .. The input and output symbols can be binary vectors of arbitrary length. Each output symbol is the sum of a randomly and independently chosen subset of the input symbols. Information describing the relations between input and output symbols is obtained at the receivers either in packet headers or by other applicationdependent means of synchronization between sender and receivers. A reliable decoding algorithm for a fountain code is one which can recover the original k input symbols from any set of m output symbols with error probability at most inversely polynomial in k. The ratio m/k is called the overhead. The encoding cost is the expected number of arithmetic operations sufficient for generating each output symbol. The decoding cost is the expected number of arithmetic operations sufficient to recover the k input symbols, divided by k. Desirable properties for a fountain code are an overhead close to 1, and constant encoding and decoding cost. A universal fountain code is one for which the decoder can recover with high probability the original symbols from any set of output symbols whose size is close to optimal.