1. Venkat Guruswami, Nicolas Resch and Chaoping Xing
Lossless Dimension Expanders via Linearized Polynomials and Subspace Designs
Venkat Guruswami, Nicolas Resch and Chaoping Xing

2. Algebraic Pseudorandomness
Traditional pseudorandom objects (e.g., expander graphs, randomness extractors, pseudorandom generators, list-decodable codes etc.) are largely combinatorial objects.
Algebraic pseudorandom objects have recently been studied. Here, dimension of subspaces corresponds to subset size.
Examples include dimension expanders, subspace designs, subspace- evasive sets, rank-preserving condensers, list-decodable rank-metric codes.
Applications include constructions of Ramsey graphs [PR’04], list- decodable codes [GX’12, GX’13, GW’14], affine extractors [Gab’11], polynomial identity testing [KS’11, FS’12].

3. Dimension Expander dim 𝑗=1 𝑑 Γ 𝑗 𝑈 ≥𝛽 dim 𝑈 .
An 𝜂,𝛽 -dimension expander is a collection of linear maps Γ 1 ,…, Γ 𝑑 : 𝑭 𝑛 → 𝑭 𝑛 such that, for any 𝑈⊆ 𝑭 𝑛 of dimension at most 𝜂𝑛, we have
dim 𝑗=1 𝑑 Γ 𝑗 𝑈 ≥𝛽 dim 𝑈 .
The degree is 𝑑.
For 𝜂𝛽<1 and 𝑑=𝑂(1), a random collection of maps will be a dimension expander with good probability; goal is to obtain an explicit construction.

4. 𝑭 𝑛
𝑭 𝑛
Γ 1 (𝑣)
Γ 2 𝑣 𝑣 𝑈
𝑗 Γ 𝑗 (𝑈)
Γ 3 (𝑣)
Γ 4 (𝑣)

5. History Wigderson defined problem in 2004. Authors Parameters
Field Restriction
Lubotzky-Zelmanov, Harrow ‘08
1 2 ,1+Ω 1
𝑹, 𝑪
Bourgain-Yehudayoff ‘13
None
Forbes-Guruswami ‘15
1 Ω 𝑑 ,Ω 𝑑
𝑭 ≥Ω( 𝑛 2 )

6. Our results
𝑑=𝑂 1 𝜖 3 , whereas random gets 𝑑=𝑂 1 𝜖 2 .
Theorem. For any 𝜖>0, there exists 𝑑=𝑑(𝜖) such that there is an explicit 1−𝜖 𝑑 , 1−𝜖 𝑑 -dimension expander of degree 𝑑 over 𝑭 𝑛 when 𝑭 ≥Ω(𝑛).
Since dim⁡( 𝑗=1 𝑑 Γ 𝑗 (𝑈))≤𝑑 dim⁡𝑈 , 𝛽= 1−𝜖 𝑑 is optimal. We call the expander lossless.
Also, 𝜂= 1−𝜖 𝑑 is optimal.
Theorem. For any 𝛿>0, there exists 𝑑=𝑑(𝛿) such that there is an explicit 1 Ω(𝛿𝑑) , Ω(𝛿𝑑) -dimension expander of degree 𝑑 over 𝑭 𝑛 when 𝑭 ≥Ω 𝑛 𝛿 .

7. Linearized Polynomials
A linearized polynomial is a polynomial in 𝑭 𝑞 𝑛 [𝑋] of the form
𝑓 𝑋 = 𝑖=0 𝑘−1 𝑓 𝑖 𝑋 𝑞 𝑖
Max 𝑖 such that 𝑓 𝑖 ≠0 is the 𝑞-degree of 𝑓(𝑋).
Denote set of all linearized polynomials of 𝑞-degree <𝑘 by 𝑭 𝑞 𝑛 𝑋; ⋅ 𝑞 <𝑘 .
If 𝛼,𝛽∈ 𝑭 𝑞 𝒏 and 𝑎,𝑏∈ 𝑭 𝑞 , then 𝑓 𝑎𝛼+𝑏𝛽 =𝑎𝑓 𝛼 +𝑏𝑓(𝛽). That is, as a map 𝑭 𝑞 𝑛 → 𝑭 𝑞 𝑛 , 𝑓 is 𝑭 𝑞 -linear.
Fact. If 𝑓∈ 𝑭 𝑞 𝑛 𝑋; ⋅ 𝑞 <𝑘 ∖{0}, then dim 𝑭 𝑞 ( ker 𝑓) ≤𝑘−1.

8. The Construction Γ 𝑗 :𝐿→ 𝑭 𝑞 𝑛 by 𝑓↦𝑓( 𝛼 𝑗 ) .
Fix 𝛼 1 ,…, 𝛼 𝑑 , a basis for 𝑭 ℎ / 𝑭 𝑞 , where ℎ= 𝑞 𝑑 and 𝑑 is degree.
Fix 𝐿⊆ 𝑭 𝑞 𝑛 𝑋; ⋅ 𝑞 <𝑘 of 𝑭 𝑞 -dimension 𝑛.
For 𝑗=1,…,𝑑, define
Γ 𝑗 :𝐿→ 𝑭 𝑞 𝑛 by 𝑓↦𝑓( 𝛼 𝑗 ) .
𝑭 𝑞 𝑛 𝑚
Intuitively, we’d like to show that subspaces of dimension 𝑠 are expanded to subspaces of dimension 𝑑−𝑘+1 𝑠.
𝑭 ℎ 𝑑
𝑭 𝑞

9. Contrapositive Characterization
Proposition. Let Γ 1 ,…, Γ 𝑑 : 𝑭 𝑛 → 𝑭 𝑛 . Suppose that ∀ 𝑉⊆ 𝑭 𝑛 s.t. dim 𝑉 ≤𝜂𝛽𝑛, we have
dim 𝑢∈ 𝑭 𝑛 :∀ 𝑗∈ 𝑑 , Γ 𝑗 𝑢 ∈𝑉 ≤ 1 𝛽 dim 𝑉 .
Then { Γ 𝑗 :𝑗∈[𝑑]} forms a (𝜂,𝛽)-dimension expander.
Thus, for 𝑉⊆ 𝑭 𝑞 𝑛 , we need to understand
𝑓∈𝐿:∀𝑗∈ 𝑑 , 𝑓 𝛼 𝑗 ∈𝑉 = 𝑓∈𝐿:𝑓 𝑭 ℎ ⊆𝑉 .
First, let us study
𝑓∈ 𝑭 𝑞 𝑛 𝑋; ⋅ 𝑞 :𝑓 𝑭 ℎ ⊆𝑉 .

10. Coding Theory Connection
The condition we study is like a “rank-metric list-recovery” problem.
[Guruswami-Wang-Xing ‘16] recently provided an explicit construction of a rank-metric code list-decodable up to the Singleton bound.
Our construction is similar, but the parameter regime is sufficiently different that we require novel constructions (particularly, the subspace design).
This is very much akin to [Guruswami-Umans-Vadhan ‘08], where an explicit construction of a bipartite expander graphs were obtained from Parvaresh-Vardy codes.

11. 𝑄 𝑌 0 , 𝑌 1 ,…, 𝑌 𝑠−1 = 𝐴 0 𝑌 0 + 𝐴 1 𝑌 1 +…+ 𝐴 𝑠−1 ( 𝑌 𝑠−1 ) ,
Interpolation
Recall: we want to understand
𝑓∈ 𝑭 𝑞 𝑋; ⋅ 𝑞 <𝑘 :𝑓 𝑭 ℎ ⊆𝑉 .
For integers 𝐷, 𝑠≤𝑚, consider
𝑄 𝑌 0 , 𝑌 1 ,…, 𝑌 𝑠−1 = 𝐴 0 𝑌 0 + 𝐴 1 𝑌 1 +…+ 𝐴 𝑠−1 ( 𝑌 𝑠−1 ) ,
each 𝐴 𝑖 𝑌 𝑖 ∈ 𝑭 𝑞 𝑛 𝑌 𝑖 ; ⋅ 𝑞 <𝐷 .
Let 𝐵≔ dim 𝑭 𝑞 𝑉 and suppose 𝐷𝑠>𝐵. Then there exists 𝑄≠0 as above such that
∀𝑣∈𝑉, 𝑄 𝑣, 𝑣 ℎ ,…, 𝑣 ℎ 𝑠−1 =0 .

12. Interpolation (cont’d)
Suppose 𝑓 𝑭 ℎ ⊆𝑉. Then, for any 𝛼∈ 𝑭 ℎ :
(𝑓 𝛼 ) ℎ = 𝑖=0 𝑘−1 𝑓 𝑖 𝛼 𝑞 𝑖 ℎ = 𝑖=0 𝑘−1 𝑓 𝑖 ℎ 𝛼 ℎ 𝑞 𝑖 = 𝑖=0 𝑘−1 𝑓 𝑖 ℎ 𝛼 𝑞 𝑖 =: 𝑓 𝜎 𝛼 ,
where 𝜎= ⋅ ℎ .
By iterating: 𝑓 𝛼 ℎ 𝑖 = 𝑓 𝜎 𝑖 𝛼 .
So,
0=𝑄 𝑓 𝛼 ,𝑓 𝛼 ℎ ,…,𝑓 𝛼 ℎ 𝑠−1 =𝑄 𝑓, 𝑓 𝜎 ,…, 𝑓 𝜎 𝑠−1 𝛼 .
If 𝐷≤𝑑−𝑘+1, then 𝑄 𝑓, 𝑓 𝜎 ,…, 𝑓 𝜎 𝑠−1 𝑋 =0.

13. Periodic Subspace Structure
The space of 𝑓 𝑋 = 𝑖 𝑓 𝑖 𝑋 𝑞 𝑖 ∈ 𝑭 𝑞 𝑛 [𝑋; ⋅ 𝑞 ] such that 𝑄 𝑓, 𝑓 𝜎 ,…, 𝑓 𝜎 𝑠−1 𝑋 =0 has the following structure:
There is an 𝑭 ℎ -subspace 𝑊⊆ 𝑭 𝑞 𝑛 of dimension ≤𝑠−1, and 𝑓 0 ∈𝑊.
There are 𝜃 1 ,…, 𝜃 𝑘−1 ∈ 𝑭 𝑞 𝑛 such that 𝑓 𝑖 ∈ 𝜃 𝑖 +𝑊.
We call such a subspace of 𝑭 𝑞 𝑛 𝑘 (𝑠−1,𝑑)-periodic subspace.
Thus, we would like to choose 𝐿 so as to have small intersection with any periodic subspace.
We will define
𝐿= 𝑓 𝑋 = 𝑖=0 𝑘−1 𝑓 𝑖 𝑋 𝑞 𝑖 : 𝑓 𝑖 ∈ 𝐻 𝑖 ∀𝑖 ,
where 𝐻 0 , 𝐻 1 ,…, 𝐻 𝑘−1 ⊆ 𝑭 𝑞 𝑛 form a subspace design.

14. Subspace Designs & Main Theorem
Definition. A collection 𝐻 0 , 𝐻 1 ,…, 𝐻 𝑘−1 ⊆ 𝑭 𝑞 𝑛 of 𝑭 𝑞 -subspaces is called a (𝑠,𝐴,𝑑)-subspace design if, for every 𝑭 𝑞 𝑑 -subspace 𝑊⊆ 𝑭 𝑞 𝑛 with dim 𝑭 𝑞 𝑑 𝑊 =𝑠,
𝑖=0 𝑘−1 dim 𝑭 𝑞 𝐻 𝑖 ∩𝑊 ≤𝐴𝑠 .
Theorem. Let 𝐻 0 , 𝐻 1 , …, 𝐻 𝑘−1 give a (𝑠,𝐴,𝑑)-subspace design for all 𝑠≤𝜇𝑛, where 0≤𝜇<1/𝑑, and assume each dim 𝑭 𝑞 𝐻 𝑖 =𝑛/𝑘. Then Γ 1 ,…, Γ 𝑑 :𝐿→ 𝑭 𝑞 𝑛 gives a 𝜇𝐴, 𝑑−𝑘+1 𝐴 -dimension expander.

15. Constructions of Subspace Designs
Theorem. [GK’16] Suppose 𝑠≤𝑡≤𝑛<𝑞. There exists a collection of 𝑀≥ 𝑞 2 4𝑡 subspaces 𝑉 1 ,…, 𝑉 𝑀 ⊆ 𝑭 𝑞 𝑛 , each of codimension 2𝑡, which forms an 𝑠, 𝑚−1 2 𝑡−𝑠+1 ,1 -subspace design.
Construction 1. Let 𝛿>0. If 𝑞≥ 𝑛 𝛿 there exists a 𝑠, 8 𝛿 , 𝑑 -subspace design for all 𝑠≤ 1−2𝛿 4𝑑 𝑛. Moreover each subspace has 𝑭 𝑞 -dimension 𝑛/𝑘.
Construction 2. Let 𝜖>0. If 𝑞≥𝑛/𝑑 there exists a 𝑠,1+𝜖, 𝑑 -subspace design for all 𝑠≤ 1−𝜖 𝑑 𝑛. Moreover each subspace has 𝑭 𝑞 -dimension 𝑛/𝑘 and 𝑘=𝑂 1 𝛿 2 , 𝑑=𝑂 1 𝛿 3 .

16. “High-degree” Folded Reed-Solomon Construction
Take 𝑉 1 ,…, 𝑉 𝑘 ⊆ 𝑭 𝑞 𝑋 <𝜖𝑛 , subspace design from [GK ’16] with 𝜖≈1/√𝑘.
Let 𝜏: 𝑭 𝑞 𝑥 → 𝑭 𝑞 𝑥 the field automorphism mapping 𝑋↦𝜁𝑋, where 𝜁 is a generator for 𝑭 𝑞 ∗ .
Define the map 𝜋: 𝑭 𝑞 𝑋 <𝜖𝑛 → 𝑭 𝑞 𝑑 𝑚 by
𝑓↦ 𝑓 𝑃 , 𝑓 𝑃 𝜏 ,…,𝑓 𝑃 𝜏 𝑚−
𝑃 is an irreducible polynomial of degree 𝑑 such that 𝑃 𝑋 , 𝑃 𝜏 =𝑃 𝜁𝑋 ,…, 𝑃 𝜏 𝑚−1 = 𝑃( 𝜁 𝑚−1 𝑋) are pairwise coprime,
𝑓 𝑃 𝜏 𝑗 is the residue of 𝑓 when evaluated at the place 𝑃.
Define 𝐻 𝑖 =𝜋 𝑉 𝑖 for 𝑖=1,…,𝑘.

17. Summary and Open Problems
Explicit construction of a 1−𝜖 𝑑 , 1−𝜖 𝑑 -dimension expander when 𝑭 ≥ Ω(𝑛), or 1 Ω 𝛿𝑑 , Ω 𝛿𝑑 -dimension expander when 𝑭 ≥ 𝑛 𝛿 .
Main ingredients: linearized polynomials, subspace designs.
Decrease the field size? Or obtain same result over 𝑹,𝑪?
Applications of dimension expanders?
Thank You!