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Wet paper codes were previously proposed as a tool for construction of steganographic schemes with arbitrary (nonshared) selection channels. In this paper, we propose a new approach to wet paper codes using random linear codes of small codimension that at the same time improves the embedding efficiency (number of random message bits embedded per embedding change). Practical algorithms are given and their performance is evaluated experimentally and compared to theoretically achievable bounds. An approximate formula for the embedding efficiency of the proposed scheme is derived. The proposed coding method can be modularly combined with most steganographic schemes to improve their security.