Performance of Hamming Codes

Evaluation of the performance of error-correcting codes has, in the past, been severely hampered by the lack of functional relationships between the uncoded and coded binit error rates. In this paper, such relationships yielding the exact decoder output error rates are developed for Hamming SED codes of lengths n = 2m - 1, m = 1, 2, 3,···, and for Hamming SEC/DED codes of length n = 2m, m = 1, 2, 3,···. In addition, for the DED codes, a similar family of formulas are derived for the probability that a received information binit is contained in a word containing an error pattern that can be detected but not corrected. A criterion of merit for the coded versus uncoded systems is postulated. A similar criterion is developed based upon word error rates. It is demonstrated that the latter results, in general, in highly erroneous conclusions regarding the comparative worth of coded systems. Graphs are presented illustrating the numerical results based on these formulas for codes ranging in length from 7/8 binits up to and including 511/512 binits, for uncoded channel error rates of 0.5 to 10-10. Similar graphs of code merit, based upon modulation systems for which detection is a linear operation (PSK-MF, for example), are constructed. From these, ranges of channel (uncoded) error probability over which particular code lengths result in the best performance that can be obtained from that type of code are extracted and tabulated.