Bounds on 2-query Locally Testable Codes with affine tests

We study Locally Testable Codes (LTCs) that can be tested by making two queries to the tested word using an a ne test. That is, we consider LTCs over a nite eld F, with codeword testers that only use tests of the form avi + bvj = c, where v is the tested word and a, b, c ∈ F. We show that such LTCs, with high minimal distance, must be of constant size. Speci cally, we show that every 2-query LTC with a ne tests over F, that has minimal distance at least 9 10 , completeness at least 1 − , and soundness at most 1− 3 , is of size at most |F|. Our main motivation in studying LTCs with a ne tests is the Unique Games Conjecture (UGC ), and the close connection between LTCs and PCPs. We mention that all known PCP constructions use LTCs with corresponding properties as building blocks, and that many of the LTCs used in PCP constructions are a ne. Furthermore, the UGC was shown to be equivalent to the UGC with a ne tests [13], thus the UGC implies the existence of a low-error 2-query PCP with a ne tests.