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.