Lossless dimension expanders via linearized polynomials and subspace designs

Abstract

For a vector space Fn over a field F, an (η, β)-dimension expander of degree d is a collection of d linear maps Γj : Fn → Fn such that for every subspace U of Fn of dimension at most ηn, the image of U under all the maps, Σj=1dΓj(U), has dimension at least β dim(U). Over a finite field, a random collection of d = O(1) maps Γj offers excellent expansion whp: β ≈ d for η ≥ ω(1/d). When it comes to a family of explicit constructions (for growing n), however, achieving even modest expansion factor β = 1 + ϵ with constant degree is a non-trivial goal.We present an explicit construction of dimension expanders over finite fields based on linearized polynomials and subspace designs, drawing inspiration from recent progress on list-decoding in the rank-metric. Our approach yields the following:• Lossless expansion over large fields; more precisely β ≥ (1 − ϵ)d and [EQUATION] with d = Oϵ(1), when |F| ≥ ω(n).• Optimal up to constant factors expansion over fields of arbitrarily small polynomial size; more precisely β ≥ ω(δd) and η ≥ ω(1/(δd)) with d = Oδ(1), when |F| ≥ nδ.Previously, an approach reducing to monotone expanders (a form of vertex expansion that is highly non-trivial to establish) gave (ω(1), 1 + ω(1))-dimension expanders of constant degree over all fields. An approach based on rank condensing via subspace designs led to dimension expanders with [EQUATION] over large fields. Ours is the first construction to achieve lossless dimension expansion, or even expansion proportional to the degree.