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Robust Codes from Higher Dimesional Expanders

Mid-Term 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 graph. Thus, high dimensional expander could be useful in applications where expander graphs could not yield strong enough results. Most notably we are aiming 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 behavior 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 expander; 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 is sure to have other applications. The resulting combinatorial objects are new tools that will probably also have powerful applications in the future. While some algorithmic applications are already known, the fact that such objects were previously not known to exist means that they will not have "off-the-shelf" applications; rather, the community now needs to revisit many research directions while taking advantage of the new opportunity that our construction provides.

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 aim to 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).

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

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 are candidate to yield better probabilistically checkable proofs that where ever known.

To conclude we are 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, 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.