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A host of emerging applications in multimedia and communications are connected to the problem of channel coding with side information (CCSI). In CCSI, side information about the transmission channel is present at the sender but not at the receiver. The goal of this paper is to bridge the gap between practical code constructions and the theoretical capacity bounds. Specifically, we present a new turbo-coded trellis-based code construction that appears to be very promising: for the Gaussian CCSI problem at a transmission rate of 1 bit/channel use, our proposed approach comes within 2.72 dB of the information-theoretic capacity established by M. Costa (see IEEE Trans. on Information Theory, vol.29, p.439-41, 1983).