An Optimal Two-Step Decoding for PSK-Modulated Noisy Index Coding

Abstract-This paper studies noisy index coding problems over broadcast channels. The codewords from a chosen binary index code of length
N
are mapped to a 2
^N
-PSK constellation before being transmitted over an AWGN channel. The receivers follow the two-step decoding process of first estimating the PSK symbol using a maximum-likelihood decoder and then performing index code decoding. After estimating the PSK symbol, there is, in general, more than one decoding strategy at a receiver, i.e., more than one linear combination of index-coded bits along with a subset of side information bits, that can be used to estimate the requested message. Thomas et al. in [“Single Uniprior Index Coding With Min–Max Probability of Error Over Fading Channels,” IEEE Transactions on Vehicular Technology, pp. 6050-6059, July 2017] showed that for binary-modulated index code transmissions, minimizing the number of transmissions used to decode a requested message is equivalent to reducing the probability of error. This paper shows that this is no longer true while employing multi-level modulations. Further, we consider the side information available to each receiver also to be noisy and derive an expression for the probability that a requested message bit is estimated erroneously at a receiver. We also show that the criterion for choosing a decoding strategy that gives the best probability of error performance at a receiver changes with the signal-to-noise ratio at which the side information is broadcast. Hence, for a given index coding problem and a chosen index code, we give an algorithm to select the best decoding strategy for the receivers. The above results are shown to be valid over fading channels also.