Given (the table of) a function f: F m → F over a finite field F, a low degree tester tests its agreement with an m-variate polynomial of total degree at most d over F. The tester is usually given access to an oracle A providing the supposed restrictions of f to affine subspaces of constant dimension (e.g., lines, planes, etc.). The tester makes very few (probabilistic) queries to f and to A (say, one query to f and one query to A), and decides whether to accept or reject based on the replies. We wish to minimize two parameters of the tester: its error and its size. The error bounds the probability that the tester accepts although the function is far from a low degree polynomial. The size is the number of bits required to write the oracle replies on all possible tester’s queries. Low degree testing is a central ingredient in most constructions of probabilistically checkable proofs (P CP s). The error of the low degree tester is related to the error of the P CP and its size is related to the size of the P CP. We design and analyze new low degree testers that have both sub-constant error o(1) and almost-linear size n 1+o(1) (where n = |F | m). Previous constructions of sub-constant error testers had polynomial size (works by Arora and Sudan  and by Raz and Safra ). These testers enabled the construction of P CP s with sub-constant error, but polynomial size (see the work by Dinur et al ). Previous constructions of almost-linear size testers obtained only constant error (Ben-Sasson, Sudan, Vadhan and Wigderson ). These testers were used to construct almost-linear size P CP s with constant error (see Ben-Sasson et al ). The testers we present in this work enabled the construction of P CP s with both sub-constant error and almost-linear size (Moshkovitz and Raz )
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