Linear Codes for Broadcasting with Noisy Side Information

We consider network coding for a noiseless broadcast channel where each receiver demands a subset of messages available at the transmitter and is equipped with noisy side information in the form an erroneous version of the message symbols it demands. We view the message symbols as elements from a finite field and assume that the number of symbol errors in the noisy side information is upper bounded by a known constant. This communication problem, which we refer to as broadcasting with noisy side information (BNSI), has applications in the re-transmission phase of downlink networks. We derive a necessary and sufficient condition for a linear coding scheme to satisfy the demands of all the receivers in a given BNSI network, and show that syndrome decoding can be used at the receivers to decode the demanded messages from the received codeword and the available noisy side information. We represent BNSI problems as bipartite graphs, and using this representation, classify the family of problems where linear coding provides bandwidth savings compared to uncoded transmission. We provide a simple algorithm to determine if a given BNSI network belongs to this family of problems, i.e., to identify if linear coding provides an advantage over uncoded transmission for the given BNSI problem. We provide lower bounds and upper bounds on the optimal codelength and constructions of linear coding schemes based on linear error correcting codes. For any given BNSI problem, we construct an equivalent index coding problem. A linear code is a valid scheme for a BNSI problem if and only if it is valid for the constructed index coding problem.