Robust Multiplication-Based Tests for Reed-Muller Codes

We consider the following multiplication-based tests to check if a given function f : Fq → Fq is the evaluation of a degree-d polynomial over Fq for q prime. Teste,k: Pick P1, . . . , Pk independent random degree-e polynomials and accept iff the function fP1 · · ·Pk is the evaluation of a degree-(d+ ek) polynomial. We prove the robust soundness of the above tests for large values of e, answering a question of Dinur and Guruswami (FOCS 2013). Previous soundness analyses of these tests were known only for the case when either e = 1 or k = 1. Even for the case k = 1 and e > 1, earlier soundness analyses were not robust. We also analyze a derandomized version of this test, where (for example) the polynomials P1, . . . , Pk can be the same random polynomial P . This generalizes a result of Guruswami et al. (STOC 2014). One of the key ingredients that go into the proof of this robust soundness is an extension of the standard Schwartz-Zippel lemma over general finite fields Fq, which may be of independent interest. 1998 ACM Subject Classification F.2.1 Numerical Algorithms and Problems