Linear Codes for Broadcasting with Noisy Side-Information and Different Error Thresholds

The problem of broadcasting with noisy side information (BNSI) consists of a set of receivers, each demanding a subset of messages available at a sender. The receivers already have a noisy version of their demands (due to decoding errors in previous transmissions) that can be considered as noisy side information for the retransmission phase. The maximum possible number of errors in the noisy side-information at a receiver is referred to as its error threshold. The objective is to reduce the number of transmissions in the retransmission phase by taking advantage of the knowledge of noisy side-information at the receivers, when each receiver has a different error threshold. We consider linear encoding schemes and provide a set of necessary and sufficient conditions for any linear code to be valid. We present a field-size independent coding scheme with low-complexity (only additions allowed in the coding scheme). This generalizes a coding scheme given in a prior work for the BNSI problem with equal error thresholds. We then provide a greedy algorithm to obtain a field-size independent encoding scheme with low-complexity for any BNSI problem with equal error thresholds.