# Explicit constructions of Expanders and LDPC codes

After seeing Nisim’s ISIT talk on LP Decoding of Regular LDPC Codes in Memoryless Channels, I remembered one issue that has been puzzling me for a while: if there are any explicit constructions of LDPC codes that have a provable nonzero threshold (for block-error probability, hence excluding density evolution arguments) i.e. correct a constant fraction of bit-flipping errors (other channels would be harder)– under any efficient decoder.
The question seems trivial: As is well-known,  a random $(d_v,d_c)$ regular graph will be a very high quality expander (for very small sets) and therefore a (very small) constant fraction of bit-flipping errors can be corrected by Sipser & Spielman‘s bit flipping algorithm.
Also the work by Feldman et al. LP decoding corrects a constant fraction of errors establishes correction of an adversarial set of bit-flipping errors for expander graphs under LP decoding.
But here is the catch: sure, a random graph is a good expander with high probability, but how do we check if a given graph is a good expander?
The key of course is my definition of explicit construction: I mean an algorithm that terminates in polynomial time and gives me a code that certifiably works (and I think this is relevant to practice, since at the end of the day I have to use one fixed code). The only efficient way that I know to check expansion involves spectral methods ( This is from Modern Coding theory by Richardson & Urbanke) : If we have a $(d_v,d_c)$ regular bipartite graph with $n$ variable nodes and $\frac{d_v}{ d_c} n$ check nodes, if we take $H$ to be the incidence matrix of the bipartite graph (the parity check matrix of the code), we can form $H^T H$ and look at its eigenvalues
$\lambda_1 \geq \lambda_2,\geq \lambda_3 \cdots$ and Tanner’s spectral bound on expansion is that expansion $\gamma$ is greater than
$\gamma(\alpha) \geq \frac{ d_v}{(d_v d_c - \lambda_2) \alpha + \lambda_2}$ for any size of expanding sets $\alpha$. Unfortunately this bound cannot certify expansion for any $\gamma >1/2$, which is exactly the point where it starts being useful for coding theory. Perhaps there are stronger spectral bounds that could establish more expansion, the book Spectral Graph Theory by Fan Chung Graham contains a lot of material on that point but I have not seen any such applications to coding.
So ok, lets say we do not know how to construct (or certify) that LDPCs have high expansion, how about other graph properties that will guarantee a correctable fraction of errors in polynomial time? This started when I was working with Costas on LP decoding for LDPC codes and we were always (incorrectly) assuming that random regular bipartite graphs will have $\log n$ girth with high probability. When we actually tried to find a proof for this, for example looking at the Modern Coding theory book we find that usual proofs establish a significantly weaker statement: in a random $(d_v,d_c)$ regular graph, if you start from a variable and start expanding the tree, you will not loop around after a constant number of steps with $1/poly(n)$ probability. This is what is refered to as ‘locally-tree like’. I do not know of any stronger statements but I think it can be easily shown that for any fixed cycle length, the expected number of cycles of that length is constant for regular random graphs.
The breakthrough paper by Arora, Daskalakis and Steurer, ‘Message-Passing Algorithms and Improved LP Decoding‘ establishes that regular LDPCs with $\log n$ girth will correct a constant fraction of random bit-flipping errors whp under LP decoding. But how do we find regular LDPCs with $\log n$ girth ?
After searching a little I found the recent work of Bayati et al.
Generating Random Graphs with Large Girth and Generating random Tanner-graphs with large girth that talk about the related problem of generating a graph with high girth uniformly from all such graphs (and with a given degree distribution) but as far as I understood these constructions cannot guarantee a diameter scaling like $\log n$ (but only any constant diameter). This is of course the relevant practical question but the scaling issue remains.
The only construction that I know is the one found in the appendix of Gallager’s thesis that contains a deterministic algorithm that constructs $(d_v,d_c)$ regular Tanner graphs with $\log n$ girth.
The same question is relevant for compressed sensing when we ask for sparsity in the measurement matrix: All the constructions that I know for sparse measurement matrices (that require the optimal number of measurements under Basis pursuit, e.g. see the survey of Gilbert and Indyk: Sparse Recovery using Sparse Matrices) are constructed from high $\gamma>2/3$ bipartite expander graphs. But given a fixed measurement matrix how do we certify that it is good and should be implemented in a system?
Of course one can use Message passing algorithms for compressed sensing (by Donoho, Maleki and Montanari) and obtain sparse measurement matrices with very good thresholds but under a different decoding algorithm.
The close connection of compressed sensing (under LP – Basis pursuit decoding) and channel coding (under LP decoding) gives one path for certifiable measurement matrices that are sparse):
1. The Arora, Daskalakis & Steurer paper guarantees that (3,6) regular graphs with $\log n$ girth correct a (quite high– much higher than expansion arguments) constant fraction of errors under LP decoding.
2. Gallager’s thesis appendix contains deterministic constructions of such (3,6) regular sparse matrices.
3. Our connection result establishes that if a matrix corrects a constant fraction of bit flipping errors, the same matrix used as a compressed sensing measurement matrix will recover all sparse signals (of the same support as the bit-flipping errors).
Combining (1)+(2)+(3) we obtain that: for sensing a length $n$ signal with $m=n/2$  measurements one recovers almost all sparse signals of size up to $0.05 n$ (whp over supports) and the $(0,1)$ measurement matrix has $O(n)$ non-zeros. (see our recent paper )
Conclusion: The Appendix C in Gallager’s thesis contains the best sparse compressed sensing measurement matrices known (under basis pursuit).