Random codes: minimum distances and error exponents

Minimum distances, distance distributions, and error exponents on a binary-symmetric channel (BSC) 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 NδGV(2R), where δGV(R) is the Gilbert-Varshamov (GV) relative distance at rate R, whereas a typical linear code (TLC) has minimum distance NδGV(R). Consequently, a TLC has a better error exponent on a BSC at low rates, namely, the expurgated error exponent.