Reducing Testing Affine Spaces to Testing Linearity of Functions

Abstract

For any finite field \(\mathcal{F}\) and \(k<\ell \), we consider the task of testing whether a function \(f:\mathcal{F}^\ell \rightarrow \{0,1\}\) is the indicator function of an \((\ell -k)\)-dimensional affine space. For the case of \(\mathcal{F}=\mathrm{GF}(2)\), an optimal tester for this property was presented by Parnas, Ron, and Samorodnitsky (SIDMA 2002), by mimicking the celebrated linearity tester of Blum, Luby and Rubinfeld (JCSS 1993) and its analysis. We show that the former task (i.e., testing \((\ell -k)\)-dimensional affine spaces) can be efficiently reduced to testing the linearity of a related function \(g:\mathcal{F}^\ell \rightarrow \mathcal{F}^k\). This reduction yields an almost optimal tester for affine spaces (represented by their indicator function).