# Sample spaces with small bias on neighborhoods and error-correcting communication protocols

## Abstract

We give a deterministic algorithm which, on input an integer *n*, collection *f* of subsets of {1,2,⋯, *n*} and *∈* ∃ (0,1) runs in time polynomial in *n*‖*f*‖/∈ and produces a ±1-matrix *M* with *n* columns and *m=O*(log ‖*f*‖/∈^{2}) rows with the following property: for any subset *F* ∃ *f*, the fraction of 1's in the *n*-vector obtained by coordinate-wise multiplication of the column vectors indexed by *F* is between (1−∈)/2 and (1+*∈*)/2. In the case that *f* is the set of all subsets of size at most *k, k* constant, this gives a polynomial time construction for a *k*-wise *∈*-biased sample space of size *O*(log *n/∈*^{2}), as compared to the best previous constructions of [NN90] and [AGHP91] which were, respectively, *O*(log *n/∈*^{4}) and *O*((log *n*)^{2}/*∈*^{2}). The number of rows in the construction matches the upper bound given by the probabilistic existence argument. Such constructions are of interest for derandomizing algorithms. As an application, we present a family of essentially optimal deterministic communication protocols for the problem of checking the consistency of two files.

## Keywords

Sample Space Explicit Construction Deterministic Algorithm Support Size Efficient Construction## Preview

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