Linearity testing in characteristic two

Let Dist(f,g) = Pr-u[f(u) fg(u) not equal 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) = Pr-u,Pr-v[f(u) + f(v) not equal 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)(n) and H = GF (2). In this case, the collection of linear functions describe a Hadamard code of block length 2(n) and for an arbitrary function f mapping GF(2)(n) 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 qnantity 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 efficient probabilistically checkable proofs and thence in the derivation of hardness of approximation results. In this context, improved analyses translate into better nonapproximability 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 is an element of [0, 1] we have L(x) less than or equal to Err(f) less than or equal to 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].