Binary Linear Codes of Minimum Distance 3 Let $B_n$ denote the maximum size of a binary linear code (a binary code that is closed under addition) whose codewords have length $n$ and whose minimum distance is $3$. I have been searching for the best-known lower bounds for $A(n,3)$, but I can't seem to find anything. My question is simply: What are some lower bounds for the numbers $B_n$?
The binary Hamming code with parameters $[n,k,d]=[2^m-1,2^m-m-1,3]$ is perfect (no larger code exists) and for the values of $n=2^m-1,$ $m\geq 3$ provides the lower bound $$A_{2^m-1}\geq 2^{(2^m-m-1)}$$.

So $A_7\geq 16,$ etc.

The Gilbert-Varshamov bound states that for alphabet size $q$ a prime power, there exists a linear $[n,k]$ code with minimum distance at least $d$ provided
$$q^{n-k}\geq \sum_{i=0}^{d-2} {n-1 \choose i} (q-1)^i.$$ So you can take $q=2$ and find the largest $k$ satisfying this inequality.

There is a table in Roman's Coding Theory book, which has the following information (but for all, not necessarily linear codes).
$$
\begin{array}{cl} 
n &amp; A(n,d) \\
5 &amp; 4\\ 6&amp; 8 \\ 7 &amp; 16\\ 8 &amp; 20\\ 9 &amp; 40 \\ 10 &amp; 72-79 \\ \cdots &amp; \cdots \\ 16 &amp; 2560-3276 \end{array}$$
Let $F$ be the finite field of size $2$ and let $K = F^n$. Of those $K$ subcodes of distance $3$ fix a $K$ subcode $C$ of maximum dimension. The unit $K$ codewords together generate $K$ while the non-trivial quotient code $K/C$ has a Hamel basis $Z$ made of unit $K/C$ codewords; i.e. made of $C$ cosets that contain a unit $K$ codeword.  

$Z$ has at most size $2$ or else $K/C$ would have a codeword of weight $3$ which is false. $C$ is not a hyperplane or else for distinct unit $K$ codewords $u,v$ we'd have $u+v$ in $C$ implying there exists a $C$ codeword of weight $2$ which is false.

Therefore, $Z = n - dim(C) = 2$ and $dim(C) = n - 2$ and $B_n = 2^{n-2}$.