What does the Gilbert-Varshamov bound say about random linear codes? I am trying to understand what the Gilbert-Varshamov bound means for random linear codes, that is, codes that are generated by a random generator matrix $G$ with $$\mathcal{C} = \{\, vG \mid v \in \mathbb{F}^k \,\}$$ or by a parity check matrix $H$ with $$\mathcal{C} = \{\, v \in \mathbb{F}^n \mid Hv = 0 \,\}.$$
Someone claimed that such a random code had an asymptotically good minimum distance according to the GV bound, and I am trying to figure out why that is. Which implications does the GV bound have for random linear codes?
 A: The proof of GV is by using the probabilistic method itself. 
In any case, the paper by Barg and Forney (Random Codes: Minimum Distances and Error Exponents, IEEE Trans. Information Theory:48(9)) available here should be very helpful.
You are looking at the LCE code ensemble.
The abstract of the paper is given below. Two quick definitions:
The RCE (Shannon) ensemble selects a random $(N,M)$ code by selecting $M$ $N-$tuples uniformly at random from $\{0,1\}^N$ (equivalently selecting each of $2^{NM}$ bits uniformly at random) so each possible code has probability $2^{-NM}$ of being selected. The rate $R$ is defined by $M=2^{RN}.$
The LCE ensemble is chosen by selecting the $K$ generating $N-$tuples $g_k$ by selecting each bit in each tuple uniformly at random and then letting the code be the binary linear span of these tuples (with dimension $K$ and $R=K/N$).
Abstract—Minimum distances, distance distributions, and error exponents
on a binary-symmetric channel (BSC) are given for typical codes from
Shannon’s random code ensemble (RCE) and for typical codes from a random
linear code ensemble (LCE). 
A typical random code of length $N$ and rate $R$ is
shown to have minimum distance $N\delta_{GV}(2R)$, where $\delta_{GV}( R)$ is the
Gilbert–Varshamov (GV) relative distance at rate $R,$ whereas a typical
linear code (TLC) has minimum distance $N \delta_{GV}( R)$. 
Consequently, a Typical Linear Code has a better error exponent on a BSC at low rates, namely, the expurgated error exponent.
