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, ∑ ^d_j=1 Γ_j(U), has dimension at least α dim(U). Over a finite field, a random collection of d = O(1) maps Γ_j 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 = O_ε(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 = O_δ(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.

对于字段\（\ mathbb {F} \）上的向量空间\（{\ mathbb {F} ^ n} \），度为d的（η，β）维扩展器是d个线性映射的集合\ （{\伽玛_j}：{\ mathbb {F} ^ N} \ RIGHTARROW {\ mathbb {F} ^ N} \），从而使每一子空间ü的\（{\ mathbb {F} ^ N} \）的至多尺寸ηN，图像ü所有地图下，Σ ^d _{Ĵ = 1} Γ _Ĵ（Û），具有尺寸至少α暗淡（Û）。在有限域上，d = O（1）的随机集合映射Γ _Ĵ提供出色的“无损”膨胀WHP： β听，说： d为≥η Ω（1 / d）。但是，当涉及到一系列显式结构（用于增长n）时，以恒定的程度获得适度的甚至扩展系数β = 1+ ε并不是一个不重要的目标。

我们提出了基于线性化多项式和子空间设计的有限域上维扩展器的显式构造，并从秩度量中列表解码的最新进展中汲取了灵感。我们的方法得出以下结果：

• 大面积无损扩展；更准确地说β ≥（1 - ε）d和\ - （\ ETA \ GE {{1个\ varepsilon} \超过d} \）与d = Ö _ε（1）中，当\（\左| \ mathbb {F} \ right | \ ge \ Omega \ left（n \ right）\）。

• 在任意小多项式大小的域上的最优常数因子展开；更准确地说β ≥ Ω（ΔD）和η ≥ Ω（1 /（ΔD）），与d = Ö _δ（1）中，当\（\左| \ mathbb {F} \右| \ GE {N ^ \增量} \）。

以前，一种简化为单调扩展器（一种极不容易建立的顶点扩展形式）的方法在所有场上都提供了恒定度的（Ω（1），1 + Ω（1））维扩展器。一种基于“通过子空间设计进行等级压缩”的方法导致使用\（\ beta \ mathbin {\ lower.3ex \ hbox {$ \ buildrel> \ over {\ smash {\ scriptstyle \ sim} \ vphantom {_x} } $}} \ sqrt d \）在较大的有限域上。我们的结构是第一个实现无损尺寸扩展甚至与度成比例的扩展的结构。