Abstract
For a vector space \({\mathbb{F}^n}\) over a field \(\mathbb{F}\), an (η, β)-dimension expander of degree d is a collection of d linear maps \({\Gamma _j}:{\mathbb{F}^n} \rightarrow {\mathbb{F}^n}\) such that for every subspace U of \({\mathbb{F}^n}\) of dimension at most ηn, the image of U under all the maps, ∑ dj=1 Γj(U), has dimension at least α dim(U). Over a finite field, a random collection of d = O(1) maps Γj offers excellent “lossless” 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 \(\eta \ge {{1 - \varepsilon} \over d}\) with d = Oε(1), when \(\left| \mathbb{F} \right| \ge \Omega \left(n \right)\). 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 \(\left| \mathbb{F} \right| \ge {n^\delta}\). 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 \(\beta \mathbin{\lower.3ex\hbox{$\buildrel>\over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} \sqrt d \) over large finite fields. Ours is the first construction to achieve lossless dimension expansion, or even expansion proportional to the degree.
Highlights
The field of pseudorandomness is concerned with efficiently constructing objects that share desirable properties with random objects while using no or little randomness
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
Beyond being interesting in their own rights, these algebraic pseudorandom objects have found many applications: for example, subspace-evasive sets have been used in the construction of Ramsey graphs [26] and list-decodable codes [19, 17]; subspace designs have been used to list-decode codes over the Hamming metric and the rank-metric [20, 17]; and rank-preserving condensers have been used in affine extractors [11] and polynomial identity testing [23, 9]
Summary
The field of pseudorandomness is concerned with efficiently constructing objects that share desirable properties with random objects while using no or little randomness. We indicate that unbalanced bipartite expander graphs play a key role in constructions of extractors and other Boolean pseudorandom objects In this scenario, the left partition is significantly larger than the right partition, but we still have that sufficiently small subsets U of the left partition expand significantly, with (1 − ε)d|U | neighbors in the right partition in the lossless case. The left partition is significantly larger than the right partition, but we still have that sufficiently small subsets U of the left partition expand significantly, with (1 − ε)d|U | neighbors in the right partition in the lossless case Such unbalanced expanders are closely related to randomness condensers, which preserve all or most of the min-entropy of a source while compressing its length. If the expansion factor is Ω(d) we deem the unbalanced dimension expander degree-proportional, while if the expansion factor is (1 − ε)d we deem it lossless
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