In this paper, we consider the problem of designing unique words (UWs) in single-carrier block transmission systems for the purpose of performing bandwidth efficient channel estimation. By first deriving the Cramer-Rao lower bound (CRLB) for the UW-based channel estimator, we show that designing optimal UWs reduces to a problem of constructing sequences of length Q <= N/2 that, when padded with zeros to make a length-N sequence, have good periodic autocorrelation properties. We then show that the only class of short (length-Q) sequences that, when padded with N - Q zeros, have perfect periodic autocorrelation properties is the class of Kronecker delta sequences. Due to the fact that these sequences typically suffer from a high peak-to-average power ratio (PAPR), we then formulate two optimisation procedures to find near-optimal UWs under certain constraints. Finally, we show that the sequences that are found with the proposed technique perform significantly better than Kronecker delta sequences that have been scaled to meet a given PAPR constraint. Copyright (C) 2008 John Wiley & Sons, Ltd.