Linearity Testing in Characteristic Two

Let Dist(f; g) = Pru [ f(u)6=g(u) ] denote the relative distance between functions f; g mapping from a group G to a group H , and let Dist(f) denote the minimum, over all linear functions (homomorphisms) g, of Dist(f; g). Given a function f : G ! H we let Err(f) = Pru;v [ f(u)+f(v)6=f(u+v) ] denote the rejection probability of the BLR (Blum-Luby-Rubinfeld) linearity test. Linearity testing is the study of the relationship between Err(f) and Dist(f), and in particular the study of lower bounds on Err(f) in terms of Dist(f). The case we are interested in is when the underlying groups are G=GF(2) and H=GF(2). In this case the collection of linear functions describe a Hadamard code of block length 2 and for an arbitrary function f mapping GF(2) to GF(2) the distance Dist(f) measures its distance to a Hadamard code (normalized so as to be a real number between 0 and 1). The quantity Err(f) is a parameter that is \easy to measure" and linearity testing studies the relationship of this parameter to the distance of f . The code and corresponding test are used in the construction of e cient probabilistically checkable proofs and thence in the derivation of hardness of approximation results. In this context, improved analyses translate into better non-approximability results. However, while several analyses of the relation of Err(f) to Dist(f) are known, none is tight. We present a description of the relationship between Err(f) and Dist(f) which is nearly complete in all its aspects, and entirely complete (i.e. tight) in some. In particular we present functions L;U : [0; 1]! [0; 1] such that for all x 2 [0; 1] we have L(x) Err(f) U(x) whenever Dist(f) = x, with the upper bound being tight on the whole range, and the lower bound tight on a large part of the range and close on the rest. Part of our strengthening is obtained by showing a new connection between the linearity testing problem and Fourier analysis, a connection which may be of independent interest. Our results are used by Bellare, Goldreich and Sudan to present the best known hardness results for Max-3SAT and other MaxSNP problems [7]. Index Terms | Probabilistically checkable proofs, approximation, program testing, Hadamard codes, error detection, linearity testing, MaxSNP. A preliminary version of this paper appeared in Proceedings of the 36th Symposium on Foundations of Computer Science, IEEE, 1995. y Department of Computer Science & Engineering, Mail Code 0114, University of California at San Diego, 9500 Gilman Drive, La Jolla, California 92093. [email protected]. z Research Division, IBM T.J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598, USA. [email protected]. x Department of Computer Science, Royal Institute of Technology, 10044 Stockholm, Sweden. [email protected]. Part of this work was done while the author was visiting MIT. { Depto. de Ingenier a Matem atica, Fac. de Cs. F sicas y Matem aticas, U. de Chile. [email protected]. This work was done while the author was at the Dept. of Applied Mathematics, Massachusetts Institute of Technology. Supported by AT&T Bell Laboratories PhD Scholarship, NSF Grant CCR{9503322, FONDECYT grant No. 1960849, and Fundaci on Andes. k Research Division, IBM T.J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598, USA. [email protected].