Optimum error detection codes for noiseless decision feedback

Iterative error-detection codes are applied to a decision-feedback communication system employing a noisy binary-symmetric forward channel and a noise-free feedback channel. The optimum (maximum information rate) code is shown to be one in which only a single parity check is made at each stage of the iteration. This code is referred to as the SPC code. For this code, the probability of an undetected error
P_M
tends to zero algebraically, rather than exponentially, with block length. However, the complexity of the necessary decoder grows only as log
\log 1/P_M
, a drastic improvement over the
\log 1/P_M
growth encountered in error-correction systems. The behavior of the SPC code under a memory (blocklength) constraint is also investigated, and the optimum length of each stage of the iteration is determined. The results have considerable practical import--modest memory size can be used with little adverse effect on information rate and with very small probability of undetected error. The over-all performance of the SPC code and decision feedback system is considered when the binary symmetric forward channel is imbedded in an additive Gaussian channel. The Gaussian channel bandwidth that maximizes information rate in bits per second is determined, under the constraint of iow error probability, and the efficiency of the code (ratio of information rate to infinite-bandwidth Gaussian channel capacity) is found to be 27 per cent, two or three times the efficiency obtainable with single-stage error-detection-only codes. Plots are provided of the variation in rate of specific SPC codes when the Gaussian channel is subject to slow fades.