Linear Codes for Broadcasting with Noisy Side Information: Bounds and Code Constructions

We consider the problem of communicating over noise free broadcast channels where each receiver possesses an erroneous version of the message symbols that it demands from the transmitter as side information, and the number of errors in this side information is upper bounded by a constant. This communication problem, which we refer to as broadcasting with noisy side information (BNSI), has applications in the retransmission phase of downlink networks, and to the best of our knowledge, has no known coding schemes available in the literature. In a BNSI network the transmitter can exploit the noisy side information at the receivers to reduce the number of uses of the broadcast channel. In this paper, using a known code design criterion, we analyze and construct linear coding schemes for BNSI networks. Using a representation of BNSI problems in terms of undirected bipartite graphs, we first derive lower bounds on the optimal codelength of linear codes for these problems. We then utilize the parity-check matrices of appropriately chosen linear error correcting codes to construct valid encoder matrices for BNSI problems. We further optimize this technique by partitioning a BNSI problem into multiple subproblems and applying independent linear encoders for each of these subproblems. Finally, we show that BNSI problems form a strict subset of index coding problems by proving that any given linear BNSI problem is equivalent to a scalar linear index coding problem, albeit with a considerably larger number of receivers than the given BNSI problem.