  The $k$-Even Set problem is a parameterized variant of the Minimum Distance
Problem of linear codes over $\mathbb F_2$, which can be stated as follows:
given a generator matrix $\mathbf A$ and an integer $k$, determine whether the
code generated by $\mathbf A$ has distance at most $k$, or in other words,
whether there is a nonzero vector $\mathbf{x}$ such that $\mathbf A \mathbf{x}$
has at most $k$ nonzero coordinates. The question of whether $k$-Even Set is
fixed parameter tractable (FPT) parameterized by the distance $k$ has been
repeatedly raised in literature; in fact, it is one of the few remaining open
questions from the seminal book of Downey and Fellows (1999). In this work, we
show that $k$-Even Set is W[1]-hard under randomized reductions. We also
consider the parameterized $k$-Shortest Vector Problem (SVP), in which we are
given a lattice whose basis vectors are integral and an integer $k$, and the
goal is to determine whether the norm of the shortest vector (in the $\ell_p$
norm for some fixed $p$) is at most $k$. Similar to $k$-Even Set, understanding
the complexity of this problem is also a long-standing open question in the
field of Parameterized Complexity. We show that, for any $p > 1$, $k$-SVP is
W[1]-hard to approximate (under randomized reductions) to some constant factor.
