Where can I find examples of "scalable" good LDPC codes, namely classical codes with properties [n, O(n), O(n] and an explicit prescription of how to increase the bit number n?
I am new to this field, so I do not know whether this is an obvious question or something that does not exist at all.
So, the state-of-the-art solution is attributed to Sipser and Shpilman, and the modern formulation of the idea is remarkably simple. Let $C_0$ be $\Delta$-length linear binary code for some finite $\Delta$. Now let $\{ G_{n} \}$ be a family of $\Delta$-regular graphs, such that $G_{n}$ has $n$ edges.
Now Define the Code $C_{n}$ to be all the assignments of $n$ bits over the $G_{n}$ edges such that any vertex in the graph sees on its local view (edges adjoint to it) a codeword of $C_{0}$. Consider the following:
That construction invented by Tanner, by definition, any check $C_{n}$ has weight at most the degree of the graph, namely $\Delta$. By simple linear algebra, one can show that if the rate of $C_{0}$ is greater than half, then $C_{n}$ has a positive rate.
Sipser and Shpilman proved that if the graph is an expander graph, then the code $C_{n}$ also has a linear distance. In particular, they present a simple proof that any sublinear assignment of bits (non-zero bits) on the edges implies that there is at least one vertex that sees a non-trivial local view (means at least one $1$-bit) yet that local view has not enough bits to exceeds the distance of $C_{0}$ and therefore can't be a codeword of it.
We know how to construct explicitly family of expanders graph, For example, consider Lubotzky, in which the graph is a Cayley graph.
So now it should be clear how to construct a family of good classical LDPC codes.
The construction of good quantum LDPC code family is not so different from the above, and even those that prove that the construction obtains a linear distance is hard the technical implementation is easy.
