The optimum mean-square decoding of general block codes

We are concerned with the use of linear codes in the channel coding part of a numerical data transmission system in which the performance criterion is the mean-square error. The optimum decoding rule is developed for a fixed code over the finite field GF(q). It is a mapping from the channel alphabet into the real numbers and as such combines both the error-correcting and digital-to-analog converter functions found in most systems. We employ abstract Fourier analysis on groups to determine an optimum 1—1 encoding rule to be used with the corresponding optimum decoder. A procedure for the simultaneous optimization over both rules is presented. We show that there is always a linear encoding rule which is one part of the optimum pairs of permissible encoding—decoding rules.A system which implements the optimum decoding rule is detailed. In this realization the outputs from a bank of generalized bandpass filters are arithmetically combined with the output of a normalizing lowpass filter to produce the optimum estimate in the decoder. These filters are mechanized in the Fourier domain by weighting the spectrum of the indicator function of the received word by selected values of the transforms of the channel transition probabilities. This approach uses complex-valued arithmetic operations as opposed to finite field operations usually found in decoders. We examine a method for reducing the complexity of the decoder and determine an exact expression for the performance of this type of suboptimal decoder