On the design of mean-square error channel coding systems using cyclic codes

Channel coding systems that employ linear block codes can be designed according to a mean-square error criterion. The optimum encoder and decoder pair is determined by selecting special elements in a generalized frequency domain based upon the values of key parameters called ratio weights. Systems may be designed that employ either hard or soft decision variables in the decoder. When cyclic codes are used, a minimal ideal decomposition of the code space shows that the ratio weights possess constant values on certain subsets in the frequency domain. This permits a drastic reduction in the number of ratio weights that need to be computed. For a cyclic code with minimal ideals, this design procedure narrows the search for constancy subsets of the ratio weights to subsets of the general space. Only one ratio weight needs to be computed for each constancy subset.