Performance of Error-Correcting Codes

Binary decoding of relatively short codes corrupted by Gaussian noise or pulse interference has been considered in applications for which the minimum useful information block length is unity, equal to the information bit capacity of a code, or equal to an integral multiple of the information capacity of a code. The following results pertain to applications which are further characterized by the performance criterion of minimizing error rate and transmitter power, with only minor restrictions on transmission bandwidth and none on channel efficiency (no credit given for error detection): [1] The word error probability for any group error-correcting code is never exceeded by its corresponding bit error probability (for either Gaussian noise or pulse interference). [2] Relatively short codes (10 to 50 information bits) of moderate redundancy (40 to 50 per cent) are available which can reduce bit error rates by factors of 10 to 400, or equivalently, permit 1 to 3 db reductions in transmitter power, when the coded system bit error rate is 10
^-6
(Gaussian noise). [3] These same codes can alternatively reduce word error rates by factors of 10 to 10
⁴
, or equivalently, permit reductions of 1.5 to 3.7 db in transmitter power, for average word error rates corresponding to values of the quantity
E/N_{o}
(signal energy per information bit/Gaussian noise power density) in the neighborhood of 8 db for the coded systems. [4] Under worst-case pulse interference, these codes permit 4 to 7 db reductions in transmitter power for average word error rates less than 20 per cent. (Considerably greater improvements result for the pulse interference expected.) [5] All of these codes are easily encoded and decoded.