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 and rate is shown to have minimum distance (2 ), where ( ) is the Gilbert–Varshamov (GV) relative distance at rate , whereas a typical linear code (TLC) has minimum distance ( ). Consequently, a TLC has a better error exponent on a BSC at low rates, namely, the expurgated error exponent.