Finite Field Operations Research Articles

On minimum mean-square error linear block codes when the data have q-adic weighting

We consider the use of an ( n, k ) lineastock code over a channel from the viewpoint of a mean-square error criterion. The code is over a finite field GF( q ) and the data to be transmitted have a natural q -adic association with the first z k integers. It is assumed that the transition probabilities of the channel satisfy an additive property. Optimum performance is judged by the mean-square of the overall error between input and output. On the basis of this criterion, we determine the optimum encoding rule and standard array decoding rule. Our analysis and synthesis of the coding rules employ the Fourier transforms of the channel transition probabilities. The optimum parameters of these rules are determined in the transform domain. The optimum decoder configuration for a rate k/n code resembles a bank of k generalized bandpass filters and is similar to a digital filter. It uses complex-valued arithmetic operations as opposed to finite field operations. The decoding system can efficiently employ fast transform techniques. Decoder storage requirements are also considered.