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?
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.