Rank Bounds for Design Matrices with Applications to Combinatorial Geometry and Locally Correctable Codes

TL;DR

This paper establishes lower bounds on the rank of design matrices with applications to combinatorial geometry and locally correctable codes, showing limitations on certain error-correcting codes and generalizing geometric theorems.

Contribution

It introduces a new rank bound for (q,k,t)-design matrices over characteristic zero fields and applies it to prove non-existence of certain 2-query LCCs over complex numbers and to extend combinatorial geometry results.

Findings

Proved a lower bound on the rank of (q,k,t)-design matrices.

Showed no infinite families of 2-query LCCs over the complex numbers exist.

Extended Sylvester-Gallai type theorems to high dimensions and colored point sets.

Abstract

A (q,k,t)-design matrix is an m x n matrix whose pattern of zeros/non-zeros satisfies the following design-like condition: each row has at most q non-zeros, each column has at least k non-zeros and the supports of every two columns intersect in at most t rows. We prove that the rank of any (q,k,t)-design matrix over a field of characteristic zero (or sufficiently large finite characteristic) is at least n - (qtn/2k)^2 . Using this result we derive the following applications: (1) Impossibility results for 2-query LCCs over the complex numbers: A 2-query locally correctable code (LCC) is an error correcting code in which every codeword coordinate can be recovered, probabilistically, by reading at most two other code positions. Such codes have numerous applications and constructions (with exponential encoding length) are known over finite fields of small characteristic. We show that infinite families of such linear 2-query LCCs do not exist over the complex numbers. (2) Generalization of results in combinatorial geometry: We prove a quantitative analog of the Sylvester-Gallai theorem: Let $v_1,...,v_m$ be a set of points in $\C^d$ such that for every $i \in [m]$ there exists at least $\delta m$ values of $j \in [m]$ such that the line through $v_i,v_j$ contains a third point in the set. We show that the dimension of $\{v_1,...,v_m \}$ is at most $O(1/\delta^2)$. Our results generalize to the high dimensional case (replacing lines with planes, etc.) and to the case where the points are colored (as in the Motzkin-Rabin Theorem).