Breaking the quadratic barrier for 3-LCC's over the reals

Abstract

We prove that 3-query linear locally correctable codes over the Reals of dimension d require block length n > d 2+λ for some fixed, positive λ > 0. Geometrically, this means that if n vectors in R d are such that each vector is spanned by a linear number of disjoint triples of others, then it must be that n > d 2+λ . This improves the known quadratic lower bounds (e.g. [KdW04,Woo07]). While a modest improvement, we expect that the new techniques introduced in this work will be useful for further progress on lower bounds of locally correctable and decodable codes with more than 2 queries, possibly over other fields as well.Our proof introduces several new ideas to existing lower bound techniques, several of which work over every field. At a high level, our proof has two parts, clustering and random restriction.The clustering step uses a powerful theorem of Barthe from convex geometry. It can be used (after preprocessing our LCC to be balanced), to apply a basis change (and rescaling) of the vectors, so that the resulting unit vectors become nearly isotropic. This together with the fact that any LCC must have many 'correlated' pairs of points, lets us deduce that the vectors must have a surprisingly strong geometric clustering, and hence also combinatorial clustering with respect to the spanning triples.In the restriction step, we devise a new variant of the dimension reduction technique used in previous lower bounds, which is able to take advantage of the combinatorial clustering structure above. The analysis of our random projection method reduces to a simple (weakly) random graph process, and works over any field.

Sylvester–Gallai for Arrangements of Subspaces

“…Our results can be interpreted in this framework as dimension upper bounds for 2-query LCC's in which each coordinate is replaced by a 'block' of k coordinate. Our results then show that, even under this relaxation, the dimension still cannot increase with n. The case of 3-query LCC's over the Reals is still wide open (some modest progress was made recently in [DSW14a]) and we hope that the methods developed in this work could lead to further progress on this tough problem.…”

“…The change of basis is found by generalizing a theorem of Barthe [Bar98] (see [DSW14a] for a more accessible treatment) from the one dimensional case (arrangement of points) to higher dimension. We state this result here since we believe it could be of independent interest.…”

Lower Bounds for Approximate LDCs

“…Here, even if we restrict our attention to real codes over R, there is still an exponential gap between lower and upper bounds. In a recent work, [DSW13], a subset of the current authors and Avi Wigderson proved an n > d 2+ǫ lower bound (for some positive ǫ) for a closely related notion of 2-query Locally Correctable Codes (LCCs) over R, improving upon the known quadratic bound. Originally, the proof of [DSW13] used a reduction from (exact) 3-LCCs over R to 2-query approximate LDCs (later, a different proof was found).…”

On Coset Leader Graphs of Structured Linear Codes

“…Specifically, we consider two families of locally constrained linear binary codes. Such codes have numerous applications in theoretical computer science (see [DSW14] and the references therein). With that, essentially in all the cases, there is a significant gap between the best known examples of such codes and upper bounds on their cardinality.…”