Abstract

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 2n 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 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 efficient 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.

Citations