Random codes: Minimum distances and error exponents

Minimum distances, distance distributions, and error exponents on a 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 has minimum distance Ndgv(2R), where dgv(R) is the Gilbert-Varshamov relative distance at rate R, whereas a typical linear code has minimum distance Ndgv(R). Consequently a typical linear code has a better error exponent at low rates.