Comment on TR07-090

The result of the paper can be deduced from already known results in [ST06] and [BHR05]. 1 MainAfter publishing the paper in ECCC, it has come to my attention that the result provenin the paper, lower bounds for adaptive linearity tests, can be deduced from alreadyknown results.As stated in the paper, the lower bound for non-adaptive linearity tests was provenby Samorodnitsky and Trevisan in [ST06], where they prove that the Complete GraphTest is optimal among all non-adaptive linearity tests.Ben-Sasson, Harsha and Raskhodnikov have proved in [BHR05] that any adap-tive linearity test with query complexity q, completeness c and soundness s can betransformed into a non-adaptive linearity test with the same query complexity, perfectcompleteness and soundness s = s+ 1− c. The combination of the two results showsoptimality of the Complete Graph Test among adaptive tests as well.We comment that the proof technique of the paper is simpler and more direct thanthe one used in [ST06]. We also study, like [ST06], the behavior of linearity testson quadratic functions. However, instead of analyzing the Gowers Norm of certainfunctions, we provide a more direct combinatorial proof, studying the behavior oflinearity tests on random quadratic functions. This proof technique also lets us provedirectly the lower bound also for adaptive linearity tests.A new paper relating to this will be posted in ECCC shortly. References[BHR05] E. Ben-Sasson, P. Harsha and S. Raskhodnikov. Some 3CNF properties arehard to test. SIAM Journal on Computing, 35(1):1–21, 2005. Preliminary Version inProc. of STOC 2003.[ST06] A. Samorodnitsky and L. Trevisan. Gowers uniformity, influence of variables,and PCPs. In Proceedings of the 38nd ACM symposium on Theory of Computation,pages 11-20, 2006. 1Electronic Colloquium on Computational Complexity, Comment 1 on Report No. 90 (2007)