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Robust Codes from Higher Dimesional Expanders

Final Report Summary - COHOMCODES (Robust Codes from Higher Dimesional Expanders)

The main object of our study in this project is a higher dimensional analogous of expander graphs and their applications.
The main agenda is that these objects (if exist) should capture provably stronger computational power than their one dimensional analogues, which are expander graphs. Thus, high dimensional expanders could be useful in applications where expander graphs could not yield strong enough results. Most notably we aim toward understanding and constructing robust codes/locally testable codes from high dimensional expanders. It is well known that expander graphs yield good codes but these codes are inherently not robust. High dimensional expanders (if exist) combine in them a random and a non random like behaviour that are both necessary for obtaining robust codes. A priori it was not at all clear if high dimensional expanders of bounded degree could at all exist; let alone their potential for robust codes etc.

We have found out the question of the existence of bounded degree high dimensional expanders coincide in fact with a seemingly unrelated question of Gromov about the existence of bounded degree complexes with the topological overlapping property. We have shown that high dimensional expansion implies topological overlapping.
We have obtained a breakthrough result that construct for the first time explicit bounded degree high dimensional expanders; This answers (positively) a question of Mikhail Gromov’s (Recipient of Abel Prize and Wolf Prize in mathematics) on the existence of previously unknown combinatorial objects with remarkable (and nonintuitive) properties (called toplogical overlapping property). A decade ago, Gromov asked whether such objects exist. Recently, we have succeeded in fully resolving Gromov’s question. Significantly, we did so by introducing a new local-global criterion that that has found many other applications e.g. in the study of quantum codes and markov chains.

The bounded degree high dimensional expanders that we have found are based on a number theoretic construction known as Ramanujan complexes.
Once we have proven the existence of bounded degree high dimensional expanders, we used them toward computer science applications and in particular, towards constructions of robust (locally testable) codes.

We have shown that the notion of high dimensional expansion is strongly related to local testability that was extensively studied by computer scientists. This put a gold triangle of relations between topology (topological overlapping), combinatorics (high dimensional expansion) and computer science (local testability). In particular, we have shown that high dimensional expansion implies locally testable codes, this explained previously known locally testable codes, as well as opened the door to new constructions of locally testable codes.

We then used high dimensional expanders to construct locally testable codes for the first time; Previously, codes constructed from expanders were not locally testable; The codes we have constructed are based on sub-quotient spaces as opposed to linear spaces as before.

We have further show that high dimensional expanders imply lattices with strong properties; this is the first time that lattices are contracted from expanders, and it gives for the first time a canonical method to construct good lattices.

We have shown that high dimensional expanders imply complexes with direct product properties; such complexes are candidate to yield better probabilistically checkable proofs that where ever known.

We have studied fast mixing of high order random walks on high dimensional expanders. This had a major affect on the study of Markov chains and it led to a resolution of one of the main open questions about finding efficient randomized algorithm for counting bases of matroids.

We have managed to construct quantum codes which are beyond the state of the art from high dimensional expanders. This also creates a bridge between physics, computer science and mathematics via quantum codes from high dimensional expanders paradigm. Our quantum codes constructed are the best LDPC quantum codes known to-date.

To conclude, we study high dimensional expanders; these fascinating objects interact with so many different disciplines such as topology; combinatorics; property testing; error correcting codes; probabilistically checkable proofs; probability, Markov chains, matroids, quantum error correcting codes etc. We are sure that the discoveries we have made and the ones yet to be discovered will be important milestone in the study of these topics and their inter-relations.