1. Locality in Coding Theory II: LTCs
Madhu Sudan
Harvard
April 9, 2016
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2. Skoltech: Locality in Coding Theory
Outline of this Part
Three ideas in Testing:
Tensor Products
Composition/Recursion
Symmetry (esp. affine-invariance)
April 9, 2016
Skoltech: Locality in Coding Theory

3. Part 0 - A Definition: Robust Testing
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Skoltech: Locality in Coding Theory

4. Recursive testing and Robustness
Generic test for code 𝐶⊆{𝑔:𝐷→ 𝔽 𝑞 }:
Given 𝑓:𝐷→ 𝔽 𝑞 , pick some set 𝑆⊆𝐷, and verify 𝑓 ​ 𝑆 ∈𝐵.
Test defined by distribution over 𝑆 and 𝐵
Often, 𝐵 itself an error-correcting code!
Robust analysis (informally):
𝑓 far from 𝐶⇒ usually 𝑓 ​ 𝑆 far from 𝐵!
(Stronger conclusion than just ¬ 𝑓 ​ 𝑆 ∈𝐵 )
April 9, 2016
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5. Recursive testing and Robustness-II
Formally: 𝐶 is 𝛼-robust w.r.t. 𝐵-test if
𝔼 𝑆 𝛿 𝑓 ​ 𝑆 ,𝐵 ≥𝛼⋅𝛿(𝑓,𝐶)
𝛼-robust ⇒ 𝛼-sound
𝜖-sound ⇒ 𝜖 ℓ -robust
Recursive testing:
If 𝐶 is 𝛼-robust wrt 𝐵-test and 𝐵 is (𝜖,ℓ)-LTC, then 𝐶 is 𝛼⋅𝜖,ℓ -LTC
Goal from now: Robust testing.
April 9, 2016
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6. Part 1: Tensor Product Codes
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7. Skoltech: Locality in Coding Theory
Tensor Product Codes
Given 𝐵⊆ 𝑓:𝑆→ 𝔽 𝑞 and 𝐶⊆ 𝑔:𝑇→ 𝔽 𝑞 ,
𝐵⊗𝐶≝ ℎ:𝑆×𝑇→ 𝔽 𝑞 | ∀𝑥, 𝑦, ℎ ⋅,𝑦 ∈𝐵 & ℎ 𝑥,⋅ ∈𝐶
Is 𝐵⊗𝐶 non-empty?
Miracle of linear algebra:
dim 𝐵⊗𝐶 = dim 𝐵 ⋅dim 𝐶
(Not to be dismissed lightly! Fails miserably even for additive codes.)
𝐵 ⊗𝑚 ≝𝐵⊗𝐵⊗⋯⊗𝐵
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8. Skoltech: Locality in Coding Theory
Tensor Product Codes - 2
Locality from tensors: If 𝑆 =𝑛 and 𝑁= 𝑛 𝑚 then
Block-length 𝐵 ⊗𝑚 =𝑁;
Constraints of length 𝑁 1 𝑚 (axis-parallel lines)
Candidate test: “Lines test”
Pick 𝑖∈[𝑚] and 𝑎 −𝑖 ∈ 𝑆 𝑚−1 uniformly. Verify 𝑓 ​ 𝑥 −𝑖 := 𝑎 −𝑖 ∈𝐹.
Sound? Robust? Hope 𝛼=𝛼(𝑚,𝛿 𝐵 ).
Question raised in [Ben-Sasson+S ’04].
Answered negatively by [P. Valiant ’05].
April 9, 2016
Skoltech: Locality in Coding Theory

9. Testing Tensor Products
Proposed by [Ben-Sasson+S’04]. Many strengthenings since, with final (?) word due to [Viderman ’11]
Test 1: Test 𝐵 𝑚 using 𝑚−1 -dimensional projections!
Lemma: [Viderman ‘11] ∀𝑚≥3, 𝛿, ∃𝛼>0 such that if 𝛿 𝐵 ≥𝛿 then 𝐵 𝑚 is 𝛼-robust wrt 𝐵 𝑚−1 -test.
Test 2: Test 𝐵 𝑚 using 2-dimensional projections!
Theorem: ∀𝑚,𝛿, ∃𝛽>0 s.t. if 𝛿 𝐵 ≥𝛿 then 𝐵 𝑚 is 𝛽-robust wrt 𝐵 2 -test
Proof: Let 𝛼 𝑚 be robustness of (m-1)-dimensional test from Lemma. Then 𝛽= 𝛼 𝑚 ⋅ 𝛼 𝑚−1 ⋯ 𝛼 2 . QED.
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10. Skoltech: Locality in Coding Theory
Proof of Lemma (𝑚=3)
Reinterpretation of robustness:
Pick random point 𝑥∈ 𝑆 3
Pick random plane 𝑝∋𝑥
Accept if 𝑓 𝑥 agrees with best 𝐶 2 codeword on 𝑝
Analysis:
𝑥 BAD if three planes disagree.
Line/plane BAD if contain many BAD points
Pr[test rejects | 𝑥 BAD] ≥ 1 3 ⇒ Pr[𝑥 BAD] small
𝑥 BAD ⇒ some line containing 𝑥 BAD ⇒ some plane containing 𝑥 BAD
Throw away BAD planes; Rest of 𝑆 3 is clean ⇒ consistent with some codeword of 𝐶 3
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11. Testing via Tensor Products
Starting with base code of rate R,
get Code of rate 𝑅 1 𝜖 , distance 1−𝑅 1 𝜖 of length 𝑁, testable with locality ℓ 𝑁 = 𝑁 𝜖
But central ingredient in much better constructions!
E.g., Apply [KMR-ZS’16a] and get code of Rate 𝑅, distance 1−𝑅 −𝜖, length 𝑁, locality ℓ 𝑁 = 𝑁 𝑜(1)
Also central ingredient in analyses of non-tensor codes!
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Skoltech: Locality in Coding Theory

12. Part 2: Testing & Composition
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13. Skoltech: Locality in Coding Theory
Zig-Zag Approach
Pioneered by [Reingold-Vadhan-Wigderson]
Identify two parameters 𝑎,𝑏 : (say we want to increase both).
Identify two operations:
Zig: 𝑎,𝑏 → 𝐴,𝛽 [ 𝛼≤𝑎≤𝐴 ;𝛽≤𝑏≤𝐵 ]
Zag: 𝑎,𝑏 →(𝛼,𝐵)
If we are lucky:
Zig ∘ Zag: 𝑎,𝑏 →( 𝐴 ′ , 𝐵 ′ )
Remarkably successful approach!
Expanders – [RVW, CRVW]
Logspace connectivity – [Reingold]
PCP – [Dinur]
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14. Zig-Zag LTC Construction [KMR-ZS’16b]
Zig-Operation: Tensoring
Length Squares
Robustness Reduces by O(1) factor
Distance Squares
Zag-Operation: Alon-Luby Transform
Distance Increases
Robustness decreases
Length no worse
Combination:
Distance Maintained!
Robustness smaller by constant factor
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15. Skoltech: Locality in Coding Theory
Zig-Zag LTC Theorem
[KMR-ZS’16] Theorem: For every 𝑅>0, for infinitely many 𝑛 there exists codes of length 𝑛 of rate 𝑅, distance 1 −𝑅 −𝑜(1) that are log log 𝑛 𝑛 , LTCs
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16. Part 3: Testing by Symmetries
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17. Skoltech: Locality in Coding Theory
Motivation
If you wish to test natural codes (linearity, low-degree, …) how to do it?
What if you want code to be LTC + LDC?
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Skoltech: Locality in Coding Theory

18. Testing via Symmetries
Idea/Hope:
If code designed to be “symmetric” and has local constraints, then it must be locally testable.
Actuality:
Not true completely generally
But with some mild restrictions.
Weakly true for “single-orbit properties”
Strongly true for “lifted codes”
Corollaries: Linearity testing, Low-degree testing …
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19. Skoltech: Locality in Coding Theory
Lifting Results
Lifting:
Base Code 𝐵⊆ 𝑏: 𝔽 𝑞 𝑡 → 𝔽 𝑞
𝑚-dim. Lift
𝐿 𝑚 𝐵 = 𝑓: 𝔽 𝑞 𝑚 → 𝔽 𝑞 𝑓 𝐴 ∈𝐵 ∀𝑡−dim affine 𝐴
Theorems:
∀𝑞 ∃𝜖 ∀𝐵,𝑡, 𝑚 𝐿 𝑚 𝐵 is 𝑞 𝑡 ,𝜖 -LTC
∀𝛿 ∃𝛼 ∀𝑞,𝑡, 𝑚,𝐵, if 𝛿 𝐵 ≥𝛿 then 𝐿 𝑚 𝐵 is 𝛼-robust wrt 𝐵 2𝑡 -test.
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20. Skoltech: Locality in Coding Theory
Single-Orbit Codes
ℓ-Constraint:
𝐾= 𝑆,𝑉 ; 𝑆⊆ 𝔽 𝑞 𝑚 , 𝑉⊆ 𝑏:𝑆→ 𝔽 𝑞 , 𝑆 =ℓ
𝑓: 𝔽 𝑞 𝑚 → 𝔽 𝑞 satisfies 𝐾 if 𝑓 ​ 𝑆 ∈𝑉
ℓ-single orbit: 𝐶⊆ 𝑔: 𝔽 𝑞 𝑚 → 𝔽 𝑞 is ℓ-single orbit if there exists an ℓ-constraint 𝐾 s.t.
𝑓∈𝑃⇔∀ affine 𝐴: 𝔽 𝑞 𝑚 → 𝔽 𝑞 𝑚 , 𝑓∘𝐴 satisfies 𝐾
Lifted Property is 𝑞 𝑡 -single orbit;
Single-orbit with 𝑆=subspace is “Lifted”.
∃ natural properties that are single-orbit, but not Lifted: e.g. deg ≤1:
(𝑆= 0, 𝑒 1 , 𝑒 2 , 𝑒 1 + 𝑒 2 , 𝑉= 𝑓:𝑓 𝑒 1 −𝑓 0 =𝑓 𝑒 1 + 𝑒 2 −𝑓( 𝑒 2 )
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21. Testing Single-Orbit Codes
Theorem: 𝐶⊆ 𝑓: 𝔽 𝑞 𝑚 → 𝔽 𝑞 is ℓ-single-orbit ⇒ 𝐶 is ℓ, 𝑂 ℓ −2 −LTC
Extremely clean and general statement
Unifies many of the “first” tests of properties. [BLR, GLRSW, RS, AKKLR, KR, JPRZ].
Clean simple proof.
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22. ℓ-single-orbit ⇒ ℓ-locally testable
𝐶⊆ 𝔽 𝑞 𝑚 → 𝔽 𝑞 given by 𝑆={ 𝛼 1 ,…, 𝛼 ℓ },𝑉 :
𝑃= 𝑓 ∀𝐴, (𝑓∘𝐴) 𝑆 ∈𝑉
“Self-correction” based-proof:
Fix 𝑓 s.t 𝜌≝Pr Rejecting 𝑓 small
Define 𝑔 from 𝑓 locally
Prove 𝑔 close to 𝑓
Prove 𝑔 satisfies constraint ∀𝐴
Only possible
𝑔 𝑥 = argmax 𝛽 Pr 𝐴:𝐴 𝛼 1 =𝑥 𝛽, 𝑓(𝐴 𝛼 2 ) ,…,𝑓(𝐴 𝛼 ℓ ) ∈𝑉
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23. Skoltech: Locality in Coding Theory
Analysis (contd.)
Vote 𝐴 𝑥 =𝛽 𝑠.𝑡. 𝛽,𝑓 𝐴 𝛼 2 ,…,𝑓(𝐴 𝛼 ℓ ) ∈𝑉
𝑔 𝑥 = majority 𝐴:𝐴 𝛼 1 =𝑥 Vote 𝐴 𝑥
Key Lemma:
∀𝑥, Pr 𝐴,𝐵:𝐴 𝛼 1 =𝐵 𝛼 1 =𝑥 Vote 𝐴 𝑥 = Vote 𝐵 𝑥 ≥1 −2ℓ𝜌
[BLR,GLRSW,RS,AKLLR,KR,JPRZ] Proofs: Build a miracle ℓ×ℓ matrix M:
Rows indexed by 𝐴 1 =𝐴, 𝐴 2 ,…, 𝐴 ℓ
Columns by 𝐵 1 =𝐵, 𝐵 2 ,…, 𝐵 ℓ
𝑀 𝑖𝑗 = 𝐴 𝑖 𝛼 𝑗 = 𝐵 𝑗 𝛼 𝑖 ∀𝑖,𝑗
Typical row/column random
Why does such a matrix exist?
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24. Matrix Magic explained
Wlog 𝐿 𝛼 1 ,…,𝐶( 𝛼 𝑡 ) independent; rest determined when 𝐶 random (affine). 𝑥
𝐴 𝛼 2
…𝐴( 𝛼 𝑡 ) …
𝐴 𝛼 ℓ
Determined
𝐵( 𝛼 2 ) 𝑡 ⋮
Random
𝐵( 𝛼 𝑡 ) ⋮
Overdetermined?
No! Linear
algebra!
𝐵( 𝛼 ℓ ) 𝑡
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25. Robust testing of Lifted Codes
Thm [GHS’15]: ∀𝛿 ∃𝛼 s.t. if 𝐵⊆ 𝑏: 𝔽 𝑞 𝑡 → 𝔽 𝑞 is a code of distance 𝛿 then 𝐿 𝑚 𝐵 is 𝛼-robust wrt 𝐵 2 -test.
Test – not most natural one!
Most natural: Inspect 𝑓 ​ 𝐴 for 𝑡-dim 𝐴
Our test: Inspect 𝑓 ​ 𝐴 for 2𝑡-dim 𝐴
Based on [Raz-Safra], [BenSassonS], …, [Viderman]
Need to show: ∀𝑓 𝔼 𝐴 𝛿 𝑓 ​ 𝐴 ,𝐵 ≥𝛿(𝑓, 𝐿 𝑚 𝐵 )
Not previously known even when 𝑡=1 and 𝐵= 𝑏 | deg 𝑏 ≤𝑑 with 𝑑= 1−𝜖 𝑞
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26. Robust Testing of Lifted Codes
For simplicity 𝐵⊆ 𝑏: 𝔽 𝑞 → 𝔽 𝑞 (𝑡=1).
General geometry + symmetry ⇒
Robust analysis with 𝑚=4 ⇒ All 𝑚
How to analyze robustness of the test for constant 𝑚?
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27. Tensors: Key to understanding Lifts
Insight: 𝐿 𝑚 𝐵 = ∩ 𝑇 𝑇 𝐵 ⊗𝑚
Wishful Approach:
Test verifies 𝛿 𝑓, 𝑇 𝐵 ⊗𝑚 small for random 𝑇
Maybe can combine this?
Approach Fails
𝛿 𝐴 𝑓 , 𝛿 𝐵 𝑓 small ≢ 𝛿 𝐴∩𝐵 𝑓 small 
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28. Skoltech: Locality in Coding Theory
Actual Analysis
Say testing 𝐿 4 (𝐵) by querying 2-d subspace.
Let 𝑃 𝑎 = {𝑓 | 𝑓 ​ line ∈𝐵 for all coordinate parallel
lines, and lines in direction 𝑎}
𝐿 4 (𝐵)= ∩ 𝑎 𝑃 𝑎 ;
𝑃 𝑎 not a tensor code, but modification of tensor analysis works!
∪ 𝑎 𝑃 𝑎 ⊆ 𝐵 ⊗4 is still an error-correcting code.
So 𝛿 𝑃 𝑎 𝑓 , 𝛿 𝑃 𝑏 𝑓 small ⇒ 𝛿 𝑃 𝑎 ∩ 𝑃 𝑏 𝑓 small!
Putting things together ⇒ Theorem.
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29. Skoltech: Locality in Coding Theory
Wrapping up
Covered:
LTCs and LDCs in high-rate regime.
Many new developments: Simple, but surprising!
Not Covered:
LTCs and LDCs in low-query regime:
[Yekhanin,Efremenko,Dvir-Gopi]: 3-query 𝔽 2 , 2-query 𝔽 2 𝑡
[Dinur, Meir, Viderman]: 3-query LTCs, Rate = 1/polylog n
Open:
LTCs + LDCs with ℓ 𝑛 =polylog 𝑛: Can you beat multivariate polynomials?
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30. Skoltech: Locality in Coding Theory
Thank You
April 9, 2016
Skoltech: Locality in Coding Theory