Open problem list for the Fine-grained Complexity Design program

In the Even Set problem, the input consists of a family F of subsets of a universe U and an integer k; the question is to find a nonempty set A ⊆ U of size at most k such that |A ∩ F | is even for every F ∈ F . Alternatively, the question can be stated as finding a nonzero codeword of Hamming weight at most k in a linear code over F2. The question of parameterized complexity of this problem, parameterized by k, remains open. Note that if we require the set A to be of size exactly k, or we require the intersections to be odd, the problem becomes W[1]-hard.