Sparse Random Linear Codes are Locally Decodable and Testable

1. Sparse Random Linear Codes are Locally Decodable and Testable Tali Kaufman (MIT) Joint work with Madhu Sudan (MIT)

2. Error-Correcting Codes • Code C ⊆{0,1}n - collection of vectors (codewords) of length n. • Linear Code- codewords form a linear subspace • Codeword weight – For c  C , w(c) is#non-zero’s in c. • C is • ntsparse if |C| = nt • n-ƴbiasedif n/2 – n1-ƴ w(c) n/2 + n1-ƴ(for every c  C ) • distance d if for every c  C w(c)  d

3. Local Testing / Correcting / Decoding GivenC ⊆{0,1}n , vector v, make k queries into v: k - local testing - decide if v is in C or far from every c  C. k - local correcting- if v is close to c  C, recover c(i) w.h.p. k - local decoding- if v is close to c  C, and c encodes a message m , recover m(i) w.h.p. [C = {E(m) | m {0,1}s }, E: {0,1}s → {0,1}n , s < n] Example: Hadamard Code, Linear functions.a {0,1}logn, f(x) =ai xi (k=3) - testing: f(x)+f(y)+f(x+y) =0 ? For random x,y. (k=2) - correcting: correct f(x) by f(x+y) + f(y) for a random y. (k=2) - decoding : recover a(i) by f(ei +y) + f(y) for a random y.

4. Brief History Local Correction: [Blum, Luby, Rubinfeld] In the context of Program Checking. Local Testability : [Blum,Luby,Rubinfeld] [Rubinfeld, Sudan], [Goldreich, Sudan] The core hardness of PCP. Local Decoding:[Katz, Trevisan], [Yekhanin] In the context of Private Information Retrieval (PIR) schemes. Most previous results (apart from [K, Litsyn] ) focus on specific codes obtained by their “nice” algebraic structures. This work: results for general codes based only on their density and distance.

5. Our Results Theorem (local-correction): For every t, ƴ > 0 const, If C ⊆{0,1}n is ntsparse and n-ƴbiasedthen it is k=k(t, ƴ ) local corrected. Corollary (local-decoding): For every t, ƴ > 0 const, If E: {0,1}t logn → {0,1}nis a linear map such that C = {E(m) | m {0,1}t logn } is ntsparse and n-ƴbiasedthen E is k=k(t, ƴ ) local decoded. Proof: CE = {(m,E(m))| m {0,1}t logn }is k local corrected. Theorem (local-testing): For every t, ƴ > 0 const, If C ⊆{0,1}n is ntsparse with distancen/2 – n1-ƴthen it is k=k(t, ƴ ) local tested . • Recall, C is • ntsparse if |C| = nt • n-ƴbiasedif n/2 – n1-ƴ w(c) n/2 + n1-ƴ (for every c  C ) • distance d if for every c  C w(c)  d

6. Corollaries Reproduce testability of Hadamard, dual-BCH codes. Random code - A random code C ⊆{0,1}n obtained by the linear span of a random t logn ∗ n matrix is ntsparse and O(logn/√n) biased, i.e. it is k= (t) local corrected, local decoded and local tested. Can not get denser random code: Similar random code obtained by a random(logn)2 ∗ n matrx doesn’t have such properties. There are linear subspaces of high degree polynomials that are sparse and un-biased so we can local correct, decode and test them. Example: Tr(ax^{2logn/4+1} + bx^{2logn/8+1} ) a,b  F_{2logn} Nice closure properties: Subcodes, Addition of new coordinates, removal of few coordinates.

7. Main Idea • Studyweight distribution of “dual code” and some related codes. • Weight distribution = ? • Dual code = ? • Which related codes? • How? – MacWilliams Identities + Johnson bounds

8. Weight Distribution, Duals • Weight distribution:(B0C,…,BnC) BkC- # of codewords of weight k in the code C. 0 k  n • Dual Code : C ┴ ⊆{0,1}n- vectors orthogonal to all codewords in C ⊆{0,1}n. Codeword v  C iff v ┴ C ┴: for every c’  C ┴,< v, c’ > = 0.

9. C C- i i Len n-2 Len n Len n-1 C- i,j i j Which Related Codes? • Local-Correction: Duals of C, C- i,C- i j • Local-Decoding: Same applied to C’. C’ = {(m,E(m))}. E(m): {0,1}s → {0,1}n , s < n • Local-Testing: Duals of C, and of C  v

10. Pk (0) = Krawtchouk Polynomial ~nk Pk (i) < (n-2i)k ~nk/2 0 n ~ -nk/2 n/2 -√(kn) n/2 n/2 +√(kn) ~nk n/2 –n1-γ n/2 +n1-ƴ 0 n Duals of Sparse Unbiased Codes have Many k-Weight Codewords Cis ntsparse and n-ƴbiased. BkC┴ = ? MacWilliams Transform : BkC┴ = BiCPk(i) / |C| BkC┴  [Pk (0) + n(1-ƴ) k · n t ] /|C| If k   ( t / ƴ) BkC┴ ~= Pk (0)/|C|

11. Canonical k-Tester Goal: Decide if v is in C or far from every c  C. Tester: Pick a random c’  [C ┴] k < v, c’ > = 0 accept else reject Total number of possible tests: | [C ┴] k| = BkC┴ For vC bad tests: | [C  v┴] k| = Bk[C  v]┴ Works if number of bad tests is bounded.

12. Pk (0) = ~nk Pk (i) < (n-2i)k ~nk/2 0 n n/2 –n1-γ n/2 +n1-ƴ ~ -nk/2 n/2 -√(kn) n/2 n/2 +√(kn) δn Johnson Bound Proof of Local Testing Theorem (un-biased) Reduces to show(Gap): for v at distance  from C: Bk[C  v]┴  (1- ) BkC┴ Using Macwilliams and the estimation BkC┴½ BiC vPk(i)  (1- ) Pk (0) Good:  loss C  v=C C +v Bad:  gain C is nt sparse and n-ƴ biased

13. Canonical k-Corrector Goal:Given v is -close to c  C, recover c(i) w.h.p. Corrector: Pick a random c’  [C ┴] k,i k-weight words w. 1 in i‘th coordinate. Return s  1c’ – { i }vs1c’= { i | c’i = 1} A random location in v is corrupted w.p. If for every i , every other coordinate j that the corrector considers is “random” then probability of error < k

14. C C- i i Len n Len n-1 Len n-2 C- i,j i j Proof of Self Correction Theorem Reduces to show (2-wise independence property in [C ┴] k ): For every i,j [C ┴] k , i,j / [C ┴] k , i  k/n (as if the code is random) [C ┴] k , i,( [C ┴] k , i,j ) k-weight codewords of C┴with 1 in i, (i & j) coordinates. [C ┴] k , i = [C ┴] k - [ C- i┴ ] k [C ┴] k , i,j = [C ┴] k - [ C- i┴ ] k - [ C- j┴ ] k + [ C- i j┴ ] k All involved codes are sparse and unbiased

15. Open Issues Local Correction based on distance. Obtain general k-local correction, local-decoding local testing results for denser codes. Which denser codes?

16. Thank You!!!