Random Codes: Minimum Distances and Error Components

Minimum distances, distance distributions, and error components on a binary symmetric channel are given for typical codes from Shannon's random code ensemble and for typical codes from a random linear code ensemble. A typical random code of length N and rate R is shown to have minimum distance N delta sub GV (2R), where delta sub GV (R) is the Gilbert-Varshamov relative distance at rate R, whereas a typical linear code has minimum distance N delta sub GV (R). It is shown that, as a consequence, a typical linear code has a better error exponent on a BSC at low rates, namely the expurgated error exponent.