On Extracting Common Random Bits From Correlated Sources on Large Alphabets

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Statistics Papers
Subject
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
Computer Sciences
Statistics and Probability
On Chan, Siu
Mossel, Elchanan
Neeman, Joe
Abstract

Suppose Alice and Bob receive strings X=(X1,...,Xn) and Y=(Y1,...,Yn) each uniformly random in [s]n, but so that X and Y are correlated. For each symbol i, we have that Yi=Xi with probability 1-ε and otherwise Yi 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.

2014-03-01
Journal title
IEEE Transactions on Information Theory