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For any finite field and , we consider the task of testing whether a function is the indicator function of an -dimensional affine space. For the case of , 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 -dimensional affine spaces) can be efficiently reduced to testing the linearity of a related function . This reduction yields an almost optimal tester for affine spaces (represented by their indicator function).

Recalling that Parnas, Ron, and Samorodnitsky used testing -dimensional affine spaces as the first step in a two-step procedure for testing k-monomials, we also show that the second step in their procedure can be reduced to testing whether the foregoing …