Complexity of decoders--I: Classes of decoding rules

Several classes of decoding rules are considered here including block decoding rules, tree decoding rules, and bounded-distance and minimum-distance decoding rules for binary parity-check codes. Under the assumption that these rules are implemented with combinational circuits and sequential machines constructed with AND gates, OR gates, INVERTERS, and binary memory cells, bounds are derived on their complexity. Complexity is measured by the number of logic elements and memory cells, and it is shown that minimum-distance and other decoders for parity-check codes can be realized with complexity proportional to the square of block length, although at the possible expense of a large decoding time. We examine tradeoffs between probability of error and complexity for the several classes of rules.