Satisfying Degree-d Equations over GF[2]

We study the problem where we are given a system of polynomial equations defined by multivariate polynomials over GF[2] of fixed, constant, degree d > 1 and the aim is to satisfy the maximum number of equations. A random assignment approximates this number within a factor 2−d and we prove that for any ε > 0, it is NP-hard to obtain a ratio 2−d +ε . When considering instances that are perfectly satisfiable we give a polynomial-time algorithm that finds an assignment that satisfies a fraction 21−d−21−2d of the constraints and we prove that it is NP-hard to do better by an arbitrarily small constant. The hardness results are proved in the form of inapproximability results for MAX-CSPs where the predicate in question has the desired form and we give some immediate results on approximation resistance of some predicates. ACM Classification: F.2.2 AMS Classification: 68Q17