By the use of abstract Fourier analysis on groups, the optimum mean-square-error decoding rule is developed for a fixed block code. The optimum one-to-one coding role to be used with this optimum decoder is derived, and a procedure for simultaneous optimization over both encoding and decoding rules is given. It is shown that there is a linear encoding rule which is optimum. A system which implements the optimum decoding rule is outlined. The major difference between this work and others involving coding for mean-square error is that the decoding rule developed here is a mapping from binaryn-tuples directly into the real numbers with the optimization being over all possible mappings into the real numbers. As such the system developed here replaces both the error-correction and digital-to-analog conversion components used in most numerical data transmission systems.