Abstract Given an AWGN channel, we look at the problem of designing an optimal binary uncoded communication system for transmitting blocks of binary symbols generated by a stationary source with memory modelled by a Markov chain (MC) or a hidden Markov model (HMM). The goal is to minimize the average SNR required for a given block error rate. The particular case where the binary source is memoryless with nonuniform symbol probabilities has been studied by Korn et al. [Optimal binary communication with nonequal probabilities. IEEE Trans Commun 2003;51:1435–8] [1] by optimally allocating the energies of the transmitted signals. In this paper we generalize the previous work to include the important case of sources with memory. The proposed system integrates the block sorting Burrows Wheeler Transform (BWT, [Burrows M, Wheeler D. A block sorting lossless data compression algorithm. Research report 124. Digital Systems Center, 1994]) [2] with an optimal energy allocation scheme based on the first order probabilities of the transformed symbols. Analytical expressions are derived for the energy gain obtained with the proposed system when compared either with the optimal blockwise MAP receiver or with a standard source coded system consisting of an optimal source encoder followed by an optimal uncoded binary communication system, i.e. by a symbol-by-symbol MAP detector.