A Simple Sub-Polynomial Degree Coboundary Expander

TL;DR

This paper introduces a simple combinatorial construction of high-dimensional expanders with spectral and coboundary expansion properties, avoiding complex algebraic methods and enabling applications in PCPs and coding theory.

Contribution

It provides the first straightforward combinatorial construction of sub-polynomial degree high-dimensional expanders with multiple expansion properties.

Findings

Construction is based on projections of the flags complex.

The complex is a local spectral, coboundary, and swap coboundary expander.

Results include near-linear size hypergraphs with good agreement tests and simple PCPs.

Abstract

High dimensional expanders simultaneously satisfying spectral and combinatorial (coboundary) expansion have recently played a major role in breakthroughs in PCP and coding theory, but the only known construction of such complexes is extremely involved, requiring deep algebraic number theory. In this work, we give an extremely simple combinatorial construction of a sub-polynomial degree complex based on projections of the flags complex (subspace chains) that is (i) a local spectral expander, (ii) a coboundary expander, and (iii) a swap coboundary expander. As a corollary, we also give the first near-linear size combinatorial hypergraphs with good agreement tests in the '1%' regime, and a simple PCP construction with near-linear size.