This paper studies noisy index coding problems over single-input single-output broadcast channels. The codewords from a chosen index code of length $N$ are transmitted after $2^N$-PSK modulation over an AWGN channel. In "Index Coded PSK Modulation for prioritized Receivers," the authors showed that when a length-$N$ index code is transmitted as a $2^N$-PSK symbol, the ML decoder at a receiver decodes directly to the message bit rather than following the two-step decoding process of first demodulating the PSK symbol and equivalently the index-coded bits and then doing index-decoding. In this paper, we consider unprioritized receivers and follow the two-step decoding process at the receivers. After estimating the PSK symbol using an ML decoder, at a receiver, there might be more than one decoding strategy, i.e., a linear combination of index-coded bits and different subsets 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,"] showed that for binary-modulated index code transmissions, minimizing the number of transmissions used to decode a requested message is equivalent to minimizing the probability of error. This paper shows that this is no longer the case while employing multi-level modulations. Further, we consider that the side information available to each receiver is also 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.

翻译：本文研究单输入单输出广播信道上的含噪索引编码问题。选定长度为$N$的索引码的码字经$2^N$-PSK调制后，通过AWGN信道传输。在《面向优先接收端的索引编码PSK调制》一文中，作者指出：当长度为$N$的索引码以$2^N$-PSK符号形式传输时，接收端的最大似然译码器直接对消息比特进行译码，而非遵循先解调PSK符号（等价于解调索引编码比特）再执行索引译码的两步流程。本文考虑无优先级的接收端，并采用两步译码流程。在使用最大似然估计器获取PSK符号后，接收端可能存在多种译码策略（即利用索引编码比特的线性组合及不同子集的边信息比特）来估计请求的消息比特。Thomas等人在《衰落信道中单唯一先验索引编码的最小化最大差错概率》一文中证明：对于二进制调制索引编码传输，最小化译码请求消息所使用的传输次数等价于最小化差错概率。本文表明，当采用多级调制时，这一结论不再成立。此外，我们假设各接收端可用的边信息也存在噪声，并推导了接收端请求消息比特估计出错的概率表达式。同时证明，接收端选择最优差错概率性能译码策略的准则会随边信息广播的信噪比变化。