## Statistics Papers

#### Document Type

Journal Article

#### Date of this Version

3-2014

#### Publication Source

IEEE Transactions on Information Theory

#### Volume

60

#### Issue

3

#### Start Page

1630

#### Last Page

1637

#### DOI

10.1109/TIT.2014.2301155

#### Abstract

Suppose Alice and Bob receive strings X=(X_{1},...,X_{n}) and Y=(Y_{1},...,Y_{n}) each uniformly random in [s]^{n}, but so that X and Y are correlated. For each symbol i, we have that Y_{i}=X_{i} with probability 1-ε and otherwise Y_{i} is chosen independently and uniformly from [s]. Alice and Bob wish to use their respective strings to extract a uniformly chosen common sequence from [s]^{k}, but without communicating. How well can they do? The trivial strategy of outputting the first k symbols yields an agreement probability of (1-ε+ε/s)^{k}. In a recent work by Bogdanov and Mossel, it was shown that in the binary case where s=2 and k=k(ε) is large enough then it is possible to extract k bits with a better agreement probability rate. In particular, it is possible to achieve agreement probability (kε)^{-1/2}·2^{-kε/(2(1-ε/2))} using a random construction based on Hamming balls, and this is optimal up to lower order terms. In this paper, we consider the same problem over larger alphabet sizes s and we show that the agreement probability rate changes dramatically as the alphabet grows. In particular, we show no strategy can achieve agreement probability better than (1-ε)^{k}(1+δ(s))^{k} where δ(s)→ 0 as s→∞. We also show that Hamming ball-based constructions have much lower agreement probability rate than the trivial algorithm as s→∞. Our proofs and results are intimately related to subtle properties of hypercontractive inequalities.

#### Keywords

hamming codes, probability, random processes, 1-ε probability, hamming ball-based constructions, agreement probability rate, common random bit extraction, correlated sources, hypercontractive inequalities, large alphabet size, lower order terms, random construction, trivial algorithm, correlation, information theory, joints, noise, noise measurement, protocols, upper bound, randomness extraction, hypercontractivity, symmetric channels

#### Recommended Citation

On Chan, S.,
Mossel, E.,
&
Neeman, J.
(2014).
On Extracting Common Random Bits From Correlated Sources on Large Alphabets.
*IEEE Transactions on Information Theory,*
*60*
(3),
1630-1637.
http://dx.doi.org/10.1109/TIT.2014.2301155

**Date Posted:** 27 November 2017

This document has been peer reviewed.