The theoretical basis for using the mean-square error criterion in channel coding systems has been established previously. Such a criterion leads to an unequal error-correcting system. This paper describes the design methods for translating that theory into practice. It also presents some new results concerning the performance of these types of systems. The relationship of the mean-square system to the one based on the minimum probability of error is examined. Except for very noisy channels the two decoders produce identical outputs for all correctable error patterns. However, the mean-square error decoder guarantees various levels of correcting capabilities for the more significant positions during overload conditions. The basic design technique relies upon generalized transforms that are closely affiliated with the Walsh functions. The fundamental parameters that are the key to the system design may be computed using a fast algorithm which is developed over a subspace in the generalized frequency domain. The decoder performs error correction with a code containingkinformation symbols by numerically combining the output from (k+ 1) finite impulse response (FIR) type of filters. The filter weights are directly related to the transforms of the system's transition probabilities. Simplification methods and precision requirements in the filter mechnizations are discussed.

Several standard signal processing techniques may be effectively applied in the design and mechanization of optimum encoders and decoders in error control systems that employ block error-correcting codes. Many of the approaches rely upon generalized transforms that resemble the Walsh functions. They use a fast algorithm that is developed over a subspace in the generalized frequency domain. The decoder performs error correction with a code which containskinformation symbols by numerically combining the output from (k+1) finite impulse response (FIR) type of filters. These filters use weights that are directly related to the transforms of the system's transition probabilities. Each filter weights the transforms of the received digits over (k+1) special subsets in the frequency domain.