Sub-constant error low degree test of almost-linear size

Given 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 a 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 (PCPs) and locally testable codes (LTCs). The error of the low degree tester is related to the soundness of the PCP and its size is related to the size of the PCP (or the length of the LTC). 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 [3, 16]. These testers enabled the construction of PCPs with sub-constant soundness, but polynomial size [3, 16, 9]. Previous constructions of almost-linear size testers obtained only constant error [13, 7]. These testers were used to construct almost-linear size LTCs and almost-linear size PCPs with constant soundness [13, 7, 5, 6, 8].