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The Coin Problem for Product Tests

Published:08 June 2018Publication History
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Abstract

Let Xm,ε be the distribution over m bits X1,…,Xm where the Xi are independent and each Xi equals 1 with probability (1−ε)/2 and 0 with probability (1 − ε)/2. We consider the smallest value ε* of ε such that the distributions Xm, ε and Xm, 0 can be distinguished with constant advantage by a function f : {0,1}mS, which is the product of k functions f1,f2,…, fk on disjoint inputs of n bits, where each fi : {0,1}nS and m = nk.

We prove that ε* = Θ(1/√n log k) if S = [−1,1], while ε* = Θ(1/√nk) if S is the set of unit-norm complex numbers.

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          cover image ACM Transactions on Computation Theory
          ACM Transactions on Computation Theory  Volume 10, Issue 3
          September 2018
          98 pages
          ISSN:1942-3454
          EISSN:1942-3462
          DOI:10.1145/3208319
          Issue’s Table of Contents

          Copyright © 2018 ACM

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          Association for Computing Machinery

          New York, NY, United States

          Publication History

          • Published: 8 June 2018
          • Accepted: 1 February 2018
          • Revised: 1 January 2018
          • Received: 1 May 2017
          Published in toct Volume 10, Issue 3

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