A deterministic approach to rate-compatible fountain communication

This paper considers a scenario in which a transmitter wishes to communicate n symbols (Galois field elements) to an arbitrary number of receivers. Each receiver knows some of the original n symbols, and we desire a transmission that allows each receiver to learn the entire n-symbol message from the fewest possible transmitted symbols. Specifically, we assume that receiver i knows k_i of the original n symbols (and their respective indices in the information vector). The value ki and the values of the indices are unknown to the transmitter. With the proposed rate-compatible transmission scheme, each receiver i can recover the original n symbols after receiving the first n - k_i transmitted symbols, the smallest number of symbols for which this is theoretically possible. The proposed scheme is based on the properties of maximum distance separable codes. A low complexity decoder implementation essentially performs Berlekamp-Massey erasure decoding of an affine shift of a Reed-Solomon code.