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
• nt sparse if |C| = nt
• n-ƴ biased if n/2 – n1-ƴ  w(c)  n/2 + n1-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) = 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.
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.
Our Results
Theorem (local-correction): For every t, ƴ > 0 const,
If C ⊆ {0,1}n is nt 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 nt sparse and n-ƴ biased then 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 nt sparse with distance n/2 – n1-ƴ then it is k=k(t, ƴ ) local
tested .
Recall, C is
• nt sparse if |C| = nt
• n-ƴ biased if n/2 – n1-ƴ  w(c)  n/2 + n1-ƴ (for every c  C )
• distance d if for every c  C w(c)  d
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 nt 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 unbiased 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.
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: (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.
Which Related Codes?
• Local-Correction: Duals of C, C- i, C- i j
C
i
Len n
C- i
i
Len n-1
• 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
- i,j
C
j
Len n-2
Duals of Sparse Unbiased Codes have
Many k-Weight Codewords
C is nt sparse and n-ƴ biased.
BkC┴ = ?
MacWilliams Transform : BkC┴ =  BiC Pk(i) / |C|
Bk
C┴ 
[Pk (0) +
n(1-ƴ) k ·
n
t]
/|C| ~nk
Pk (0) =
n
 
k 
Krawtchouk
Polynomial
Pk (i) < (n-2i)k
If k   ( t / ƴ)
~nk/2
0
k/2
~ -n
n/2 –n1-γ
n/2 -√(kn)
BkC┴ ~= Pk (0)/|C|
n/2 +n1-ƴ
n/2 n/2 +√(kn)
n
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.
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  v Pk(i)  (1-  ) Pk (0)
~nk
Pk (0) =  n 
k 
 
Good:
 loss
C  v = C  C +v
Pk (i) < (n-2i)k
Bad:
 gain
Johnson Bound
~nk/2
0
~ -nk/2
δn
n/2 –n1-γ
n/2 -√(kn)
n/2 +n1-ƴ
n/2
n/2 +√(kn)
n
C is nt sparse
and n-ƴ biased
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 } vs
1c’ = { 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
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
i
Len n
C- i
i
Len n-1
[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
j
C- i,j
Len n-2
Open Issues
Local Correction based on distance.
Obtain general k-local correction, local-decoding local testing
results for denser codes.
Which denser codes?
Thank You!!!