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

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 n t sparse if |C| = n t n -ƴ biased if n/2 – n 1-ƴ  w(c)  n/2 + n 1-ƴ (for every c  C ) distance d if for every c  C w(c)  d

Local Testing / Correcting / Decoding Given C ⊆ {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) =  a i x i (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(e i +y) + f(y) for a random y.

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.

Theorem (local-correction): For every t, ƴ > 0 const, If C ⊆ {0,1} n is n t sparse and n -ƴ biased then 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} n is a linear map such that C = {E(m) | m  { 0,1} t logn } is n t sparse and n -ƴ biased then E is k=k(t, ƴ ) local decoded. Proof: C E = {(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 n t sparse with distance n/2 – n 1-ƴ then it is k=k(t, ƴ ) local tested. Our Results Recall, C is n t sparse if |C| = n t n -ƴ biased if n/2 – n 1-ƴ  w(c)  n/2 + n 1-ƴ (for every c  C ) distance d if for every c  C w(c)  d

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 n t sparse 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^{2 logn/4+1 } + bx^{2 logn/8+1 } ) a,b  F_{2 logn } Nice closure properties: Subcodes, Addition of new coordinates, removal of few coordinates. Corollaries

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

Weight Distribution, Duals Weight distribution: (B 0 C,…,B n C ) B k C - # 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 ┴, = 0.

Which Related Codes? 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 C C - i i ij C - i,j Len nLen n-1 Len n-2 Local-Correction: Duals of C, C - i, C - i j

C is n t sparse and n -ƴ biased. B k C┴ = ? Duals of Sparse Unbiased Codes have Many k-Weight Codewords n ~n k 0 P k (i) < (n-2i) k Krawtchouk Polynomial n/2 ~n k/2 ~ -n k/2 n/2 -√(kn) n/2 +√(kn) P k (0) = B k C┴  [P k (0) + n (1-ƴ) k · n t ] /|C| If k   ( t / ƴ) B k C┴ ~= P k (0)/|C| n/2 –n 1-γ n/2 +n 1-ƴ n ~n k 0 MacWilliams Transform : B k C┴ =  B i C P k (i) / |C|

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

Proof of Local Testing Theorem ( un-biased ) n ~n k 0 P k (i) < (n-2i) k n/2 ~n k/2 ~ -n k/2 n/2 -√(kn) n/2 +√(kn) n/2 –n 1-γ n/2 +n 1-ƴ P k (0) = δnδn Johnson Bound Reduces to show (Gap): for v at distance  from C: B k [C  v]┴  (1-  ) B k C┴ Using Macwilliams and the estimation B k C┴ ½  B i C  v P k (i)  (1-  ) P k (0) C is n t sparse and n -ƴ biased Good:  loss Bad:  gain C  v = C  C +v

Canonical k-Corrector Corrector: Pick a random c’  [C ┴ ] k,i k-weight words w. 1 in i‘th coordinate. Return  s  1 c ’ – { i } v s 1 c ’ = { 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 Goal: Given v is  -close to c  C, recover c(i) w.h.p.

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. Proof of Self Correction Theorem C C - i i ij C - i,j Len nLen n-1Len n-2 [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

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

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