Sublinear-Time Error-Correction and Error-Detection

Sublinear-Time Error-Correction and Error-Detection Luca Trevisan U.C. Berkeley luca@eecs.berkeley.edu

Contents • Survey of results on error-correcting codes with sub-linear time checking and decoding procedures • Most of the results not proved by the speaker • Some of the results not yet proved by anybody

Error-correction

Error-detection

Minimum Distance

Ideally • Constant information rate • Linear minimum distance • Very efficient decoding Sipser-Spielman: linear time deterministic procedure

Sub-linear time decoding? • Must be probabilistic • Must have some probability of incorrect decoding • Even so, is it possible?

Reasons to be interested • Sub-linear time decoding useful for worst-case to average-case reductions, and in information-theoretic Private Information Retrieval • Sub-linear time checking arises in PCP • Useful in practice?

Hadamard Code

“Constant time” decoding

Analysis

A Lower Bound • If: the code is linear, the alphabet is small, and the decoding procedure uses two queries • Then exponential encoding length is necessary Goldreich-Trevisan, Samorodnitsky

More trade-offs • For k queries and binary alphabet: • More complicated formulas for bigger alphabet

Construction without polynomials

Negative result 1 • Suppose C:{0,1}^n -> {0,1}^m is code with decoding procedure that reads only k bits of corrupted encoding • Pick random x, compute C(x), project C(x) on m^{(k-1)/k}coordinates, prove that it still contains W(n) bits of info. about x. • Then it must be m=W(n^{k/(k-1)}) Katz-Trevisan

Negative Result 2 • Suppose C:{0,1}^n -> {0,1}^m is linear code with decoding procedure that reads only 2 bits of corrupted encoding • Then there are vectors a1…am in {0,1}^n such that for each i=1,…,n there are W(m)disjoint pairs j1,j2 such that aj1 xor aj2 = ei • Then it must be m=exp(W(n)) Goldreich-Trevisan, Samorodnitksy

Checking polynomial codes • Consider encoding with multivariate low-degree polynomials • Given p, pick random z, do the decoding for p(z), compare with actual value of p(z) • “Simple” case of low-degree test. Rejection prob. proportional to distance from code. Rubinfeld-Sudan

Bivariate Low Degree Test • A degree-d bivariate polynomial p:F x F -> F is represented as 2|F| elements of F^d (the univariate polynomial qa (y) = p(a,y) for each a and the polynomial rb(x) = p(x,b) for each b • Test: pick random a and b, read qa and rb, check that qa(b)=rb(a)

Analysis • If |F| is a constant factor bigger than d, then rejection probability is proportional to distance from code Arora-Safra, ALMSS, Polishuck-Spielman

Efficiency of Decoding vs Checking

Tensor Product Codes • Suppose we have a linear code C with codewords in {0,1}^m. • Define new code C’ with codewords in {0,1}^(mxm); • a “matrix” is a codeword of C’ if each row and each column is codeword for C • If C has lots of codeword and large minimum distance, same true for C’

Generalization of the Bivariate Low Degree Test • Suppose C has K codewords • Define code C’’ over alphabet [K], with codewords of length 2m • C’’ has as many codewords as C’ • For each codeword yof C’, corresponding codeword in C’’ contains value of each row and each column of y • Test: pick a random “row” and a random “column”, check intersection agrees • Analysis?

Negative Results? • No known lower bound for locally checkable codes • Possible to get encoding length n^(1+o(1)) and checking with O(1) queries and {0,1} alphabet? • Possible to get encoding length O(n) with O(1) queries and small alphabet?

Applications? • Better locally decodable codes have applications to PIR • General/simple analysis of checkable proofs could have application to PCP (linear-length PCP, simple proof of the PCP theorem) • Applications to the practice of fault-tolerant data storage/transmission?

Sublinear-Time Error-Correction and Error-Detection