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 affine test.
• We show that such LTCs, with high minimal distance, must be of constant size.
• Our main motivation in studying such LTCs is the Unique Games Conjecture, and the close connection between LTCs and PCPs.

We study Locally Testable Codes (LTCs  ) that can be tested by making two queries to the tested word using an affine test. That is, we consider LTCs over a finite field FF, with codeword testers that only use tests of the form avi+bvj=cavi+bvj=c, where v   is the tested word and a,b,c∈Fa,b,c∈F.We show that such LTCs, with high minimal distance, must be of constant size. Specifically, we show that every 2-query LTC with affine tests over FF, that has minimal distance at least 910, completeness at least 1−ϵ1−ϵ, and soundness at most 1−3ϵ1−3ϵ, is of size at most |F||F|.Our main motivation in studying LTCs with affine 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 affine. Furthermore, the UGC was shown to be equivalent to the UGC with affine tests [13], thus the UGC implies the existence of a low-error 2-query PCP with affine tests. We note, however, that our result has no implication on the correctness of the UGC.