Linearity testing in characteristic two

Let Dist(f,g)=Pr_u[f(u)≠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,υ[f(u)+f(υ)≠f(u+υ)] denote the rejection probability of the Blum-Luby-Rubinfeld (1993) linearity test. Linearity testing is the study of the relationship between Err(f) and Dist(f), and in particular lower bounds on Err(f) in terms of Dist(f). We discuss 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 2ⁿ and for an arbitrary function f mapping GF(2)ⁿ to GF(2) the distance Dist(l) measures its distance to a Hadamard code. 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. Improved analyses translate into better nonapproximability results. We present a description of the relationship between Err(f) and Dist(f) which is nearly complete in all its aspects, and entirely complete in some. We present functions L,U:[0,1]→[0,1] such that for all x ∈ [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 linearity testing and Fourier analysis