Affine Invariant LCCs and LTCs Sivakanth Gopi Joint work with Arnab Bhattacharya (Indian Institute of Science)
Error Correcting Code • Σ : finite alphabet, Χ : set of coordinates of size 𝑂 • Σ $ : set of all functions from Χ → Σ • Hamming distance, Δ 𝑔, = Pr -∈$ 𝑔 𝑦 ≠ 𝑦 • 𝐷 ⊂ Σ $ : Error correcting code with minimum distance 𝜀 if Δ 𝑔, ≥ 𝜀 for all 𝑔, ∈ 𝐷 6:Χ → Σ Corrupted word 𝑔 Codeword 𝑔: Χ → Σ Corruptions 6 < 𝜀 Corrector A Δ 𝑔, 𝑔 2 What if I am interested in correcting only 6? one coordinate of 𝑔
Locally Correctable Code (LCC) • Can correct any coordinate of a corrupted codeword by querying only 𝑠 locations 6:Χ → Σ Corrupted word 𝑔 Codeword 𝑔: Χ → Σ Corruptions 6 ≤ 𝜀 Δ 𝑔, 𝑔 𝑠 queries 𝑦 ∈ Χ 𝑔 𝑦 Local Corrector A w.h.p 6,𝐷 ≤ 𝜀, locally? How do we know if Δ 𝑔
Locally Testable Code (LTC) • Can test closeness to the code by querying only 𝑠 locations 6:Χ → Σ 𝑔 𝑠 queries Accepts w.p > = 6 ∈ 𝐷 > if 𝑔 Local Tester A Rejects w.p > = 6,𝐷 > 𝜀/4 > if Δ 𝑔
What’s known? • In this talk, constant query: 𝑠 = 𝑃(1) , constant alphabet: Σ = O(1) , • Let Χ = N, the length of messages we can encode is log|𝐷| Bounds on log|𝐷| Lower Bound Upper Bound log 𝑂 𝑃 log𝑂 2-query LCC [Hadamard Code] [KdW04] (log𝑂) MNO 𝑂 ONO/ M/P 𝑠 -query LCC ( 𝑠 ≥ 3 ) [Reed Muller Codes] [KT00,KdW04,Woo07] 𝑠 -query LTC ( 𝑠 ≥ 2 ) 𝑂/𝑞𝑝𝑚𝑧𝑚𝑝(𝑂) 𝑃(𝑂) [BS05,Din07] [Trivial]
Local codes from invariance • LCCs and LTCs need to satisfy many local constraints ∀𝑔 ∈ 𝐷 , Γ 𝑔 𝑦 O ,⋯ , 𝑔 𝑦 M = 1 • Let 𝐻 be a group acting on Χ and so 𝐻 also acts on functions 𝑔: Χ → Σ as 𝛿 𝑔 (𝑦) = 𝑔 ∘ 𝛿(𝑦) • Let code 𝐷 ⊂ Σ $ be invariant under this action i.e. ∀𝑔 ∈ 𝐷, 𝛿 ∈ 𝐻: 𝑔 ∘ 𝛿 ∈ 𝐷 • Local constraint on 𝑦 O , ⋯ , 𝑦 M ⇒ Local constraint on 𝛿 𝑦 O , ⋯, 𝛿 𝑦 M for all 𝛿 ∈ 𝐻 ∀𝑔 ∈ 𝐷, 𝛿 ∈ 𝐻 Γ 𝑔 𝛿(𝑦 O ) , ⋯, 𝑔 𝛿(𝑦 M ) = 1
Affine invariant codes • Kaufman and Sudan in ‘07 • 𝔾: any finite field. Let Χ = 𝔾 ] and let 𝐻 = Aff(n, 𝔾) be the group of invertible affine maps from 𝔾 ] → 𝔾 ] • A code 𝐷 ⊂ Σ 𝔾 a which is invariant under the action of Aff(𝑜, 𝔾) is called affine invariant i.e. ∀𝑔 ∈ 𝐷, ∀ℓ ∈ Aff(𝑜, 𝔾), 𝑔 ∘ ℓ ∈ 𝐷 • Example • Reed-Muller code of degree 𝑒 : set of polynomial functions of degree ≤ 𝑒 from 𝔾 ] → 𝔾 • If 𝑔 𝑦 is a degree ≤ 𝑒 polynomial and ℓ 𝑦 = 𝐵𝑦 + 𝑐 , then 𝑔 ℓ 𝑦 is also a degree ≤ 𝑒 polynomial • Irreducible polynomials, products of two degree 𝑒 polynomials... Can we construct good LCCs or LTCs using affine invariance?
Main Results Locally Correctable Codes If 𝐷 ⊂ Σ 𝔾 a is an 𝑠 -query affine invariant LCC then log 𝐷 ≤ 𝑃 M, 𝔾 , i (𝑜 MNO ) (Note that 𝑜 = log |𝔾| 𝑂 , where 𝑂 is length of the code) • Achieved by Reed-Muller codes of degree 𝑠 − 1 Locally Testable Codes If 𝐷 ⊂ Σ 𝔾 a is an 𝑠 -query affine invariant LTC then log 𝐷 ≤ 𝑃 M, 𝔾 , i (𝑜 MNP ) • Achieved by Lifted Codes of [GKS’13] • [Ben-Sasson, Sudan ‘11] proved the same bounds when Σ is a subfield of 𝔾 and 𝐷 is a linear code over Σ
Higher Order Fourier Analysis
Gowers uniformity norms ] → ℂ as • Define multiplicative derivative of 𝑔: 𝔾 j Δ v 𝑔 𝑦 = 𝑔 𝑦 + ℎ 𝑔 𝑦 ] → ℂ • Gowers uniformity norm of order 𝑒 + 1 of 𝑔: 𝔾 j O/P w | 𝑔 | p qrs = 𝔽 -,v s ,⋯,v qrs ∈𝔾 a Δ v s ⋯ Δ v qrs 𝑔 𝑦 ] → 𝔾 j is • If 𝑔 𝑦 = 𝜕 m - where 𝜕 : 𝑞 no root of unity and : 𝔾 j a degree 𝑒 polynomial then O/P w | 𝑔 | p qrs = 𝔽 -,v s ,⋯,v qrs ∈𝔾 a 𝜕 y zs ⋯y zqrs m - = 1 • Inverse Gowers theorem [Tao, Ziegler ’11]: (𝑞 > 𝑒) If | 𝑔 | p qrs = Ω 1 then 𝑔 is correlated with the phase of a degree 𝑒 polynomial • For 𝑞 ≤ 𝑒 , we get non-classical polynomials
Von Neumann inequality • If | 𝑔 | p { ≪ 1 , then cannot find 𝑔 at ℓ 𝑦 } from the values of at ℓ 𝑦 O ,⋯ ℓ 𝑦 M for a random ℓ ∈ ~ Aff(𝔾 j , 𝑜) ≤ 2 M | 𝑔 | p { 𝔽 ℓ 𝑔 ∘ ℓ 𝑦 } Γ ∘ ℓ 𝑦 O , ⋯ , ∘ ℓ 𝑦 M • Proof: expand Γ in Fourier basis, make linear change of ƒ doesn’t ‚ variables to get expressions like depend on 𝑨 ‚ ƒ −𝑨 O + ∑𝑨 ‚ ⋯ M ƒ −𝑨 M + ∑𝑨 ‚ 𝔽 • s ,⋯,• { 𝑔 ∑𝑨 ‚ O • and repeatedly apply Cauchy-Schwarz inequality ƒ −𝑨 O + ∑𝑨 ‚ ⋯ M ƒ −𝑨 M + ∑𝑨 ‚ 𝔽 • s ,⋯,• { 𝑔 ∑𝑨 ‚ ≤ | 𝑔 | p { O
Proof sketch for LCCs
Some simplifications • Assume Σ = −1,1 , 𝔾 = 𝔾 j for some prime 𝑞 > 𝑠 • Assume perfect recovery for codewords Codeword word 𝑔: Χ → Σ 𝑦 O 𝑦 P 𝑦 M Local Corrector A 𝑦 ∈ Χ Γ - s ,⋯,- { 𝑔 𝑦 O ,𝑔 𝑦 P ,⋯ , 𝑔 𝑦 M = 𝑔(𝑦) 𝑦 O ,𝑦 P ,⋯, 𝑦 M ∼ ℳ -
Proof Sketch • Step 1 : Show that any two distinct codewords 𝑔, ∈ 𝐷 must be 2 𝜗 -far in 𝑉 M -norm i.e. |𝑔 − | p { > 2𝜗 (von Neumann inequality) • Step 2 : Construct a small 𝜗 -net 𝒪 for the set of all functions in 𝑉 M -norm (Inverse Gowers theorem) • 𝒪 = {red points}, 𝐷 = {green points}, two green dots cannot fall in the same ball! • 𝐷 ≤ 𝒪 𝜗 > 2𝜗
Proof of Step 1 • Intuitively, if |𝑔 − | p { < 2𝜗 then the local corrector cannot distinguish between 𝑔 ∘ ℓ, ∘ ℓ for a random ℓ ∈ Aff 𝑜, 𝔾 j • But 𝑔 ∘ ℓ, ∘ ℓ are valid codewords by invariance and the corrector should distinguish them – Contradiction! Codeword word 𝑔 ∘ ℓ: Χ → Σ 𝑦 P 𝑦 O 𝑦 M 𝑔 ∘ ℓ(𝑦) ? Local Corrector A 𝑦 ∈ Χ Γ - s ,⋯,- { 𝑔 ∘ ℓ 𝑦 O , ⋯, 𝑔 ∘ ℓ 𝑦 M 𝑦 O ,𝑦 P ,⋯, 𝑦 M ∼ ℳ - ∘ ℓ(𝑦) ?
Proof of Step 1 - [ A ‘∘ℓ outputs 𝑔 ∘ ℓ(𝑦) ] – Pr - [ A ‘∘ℓ outputs ∘ ℓ(𝑦) ] Pr • = 1 − Pr - [𝑔 ∘ ℓ 𝑦 = ∘ ℓ(𝑦)] • = Δ 𝑔, ≥ 𝑒𝑗𝑡𝑢(𝐷) • O - [ A ‘∘ℓ outputs 𝑔 ∘ ℓ 𝑦 ] − Pr - [ A ‘∘ℓ outputs ∘ ℓ 𝑦 ] P 𝔽 ℓ Pr • 𝔽 ℓ 𝔽 - 𝔽 - s ,⋯,- { ∼ℳ ˜ 𝑔 ∘ ℓ 𝑦 − ∘ ℓ 𝑦 Γ - s ,⋯,- { 𝑔 ∘ ℓ 𝑦 O ,⋯ ,𝑔 ∘ ℓ 𝑦 M • 𝔽 - 𝔽 - s ,⋯,- { ∼ℳ ˜ 𝔽 ℓ 𝑔 ∘ ℓ 𝑦 − ∘ ℓ 𝑦 Γ - s ,⋯,- { 𝑔 ∘ ℓ 𝑦 O ,⋯ ,𝑔 ∘ ℓ 𝑦 M • ≤ 2 M 𝑔 − p { (von Neumann inequality) ™‚š› œ Therefore 𝑔 − p { ≥ 2 = 2𝜗 • P {
Proof of Step 2 (small 𝜗 -net) • Decomposition theorem ( Green, Tao, Ziegler’11 ) ] → −1,1 can be ∀𝜗, 𝑠 ∃𝑙(𝜗, 𝑠) such that: any ℎ: 𝔾 j 𝜗 -approximated by a function of 𝑙 degree 𝑠 − 1 polynomials in 𝑉 M - norm ℎ − Γ 𝑞 O , ⋯, 𝑞 p { < 𝜗 • A degree 𝑠 − 1 polynomial has 𝑜 MNO coefficients • Gives an epsilon-net of size 𝒪 = exp 𝑃 j,M 𝑜 MNO • Thus 𝐷 ≤ 𝒪 = exp 𝑃 j,M 𝑜 MNO QED!
Open Questions • We show “tight” bounds on the size of affine invariant constant query LCCs and LTCs • Improve the dependence on 𝑠, 𝔾 , |Σ| • Can we prove similar bounds for a more general class of codes? Codes invariant under some group action and some additional properties? • Can we use sparse hypergraph regularity lemmas to understand the hypergraph structure of local codes? ARIGATO GOZAIMASU!
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