# What are examples of scalable classical LDPC codes?

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.

# Good Classical LDPC Code of Infinitely Large Lengths.

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:

1. 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.

2. 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.

3. 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.

# BTW - QLDPC.

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.

• Thanks. If I require only a constant rate and $d \geq \sqrt{n}$, are there other examples? Mar 10 at 7:52