Optimum mean-square error use of convolutional codes

The
k
input and output digits of a rate
(k/n)
linear convolutional code over a finite field GF
(q)
are related to a finite set of integers by a
q
-ary expansion. The mean-square error criterion is used to simultaneously select the optimum encoder and decoder rules. This optimization is performed over all one-to-one generalized encoding rules and all decoding functions that map into the real numbers. The optimal design procedure relies upon generalized Fourier transforms, and it is shown that the encoder part of the optimum pair of rules can be taken as a linear function when the input space of symbols is viewed in a natural algebraic setting. The decoder part is a conditional mean estimator coupled with a rounding operation. One method of implementing the decoder uses the nonlinear combination of filter functions defined in the generalized frequency domain.