The non-adaptive query complexity of testing k-parities

TL;DR

This paper establishes tight bounds of Theta(k log k) queries for non-adaptive testing of k-parities, combining communication complexity techniques to determine the query complexity required.

Contribution

It provides the first tight bounds for non-adaptive testing of k-parities, integrating recent communication complexity methods with lower bounds for k-disjointness.

Findings

Proves tight Theta(k log k) query bounds for testing k-parities.

Uses communication complexity to derive lower bounds.

Connects testing complexity with k-disjointness communication bounds.

Abstract

We prove tight bounds of Theta(k log k) queries for non-adaptively testing whether a function f:{0,1}^n -> {0,1} is a k-parity or far from any k-parity. The lower bound combines a recent method of Blais, Brody and Matulef [BBM11] to get lower bounds for testing from communication complexity with an Omega(k \log k) lower bound for the one-way communication complexity of k-disjointness.