High dimensional expansion implies amplified local testability

TL;DR

This paper demonstrates that high dimensional expansion directly implies amplified local testability of codes, providing a unified framework that strengthens existing results for affine invariant codes.

Contribution

It introduces the concept of high-dimensional-expanding-systems (HDE-systems) and proves that codes modeled over them are amplified locally testable, unifying and strengthening prior local testability results.

Findings

High-dimensional expansion implies amplified local testability.

Most well-studied locally testable codes fit into the HDE-system framework.

The framework yields the strongest known testing results for affine invariant codes.

Abstract

In this work we show that high dimensional expansion implies locally testable code. Specifically, we define a notion that we call high-dimensional-expanding-system (HDE-system). This is a set system defined by incidence relations with certain high dimensional expansion relations between its sets. We say that a linear code is modelled over HDE-system, if the collection of linear constraints that the code satisfies could by described via the HDE-system. We show that a code that can be modelled over HDE-system is locally testable. This implies that high dimensional expansion phenomenon solely implies local testability of codes. Prior work had to rely to local notions of local testability to get some global forms of testability (e.g. co-systolic expansion from local one, global agreement from local one), while our work infers global testability directly from high dimensional expansion without relying on some local form of testability. The local testability result that we obtain from HDE-systems is, in fact, stronger than standard one, and we term it amplified local testability. We further show that most of the well studied locally testable codes as Reed-Muller codes and more generally affine invariant codes with single-orbit property fall into our framework. Namely, it is possible to show that they are modelled over an HDE-system, and hence the family of all p-ary affine invariant codes is amplified locally testable. This yields the strongest known testing results for affine invariant codes with single orbit, strengthening the work of Kaufman and Sudan.