  A functional $k$-PIR code of dimension $s$ consists of $n$ servers storing
linear combinations of $s$ linearly independent information symbols. Any linear
combination of the $s$ information symbols can be recovered by $k$ disjoint
subsets of servers. The goal is to find the smallest number of servers for
given $k$ and $s$. We provide lower bounds on the number of servers and
constructions which yield upper bounds on this number. For $k \leq 4$, exact
bounds on the number of servers are proved. Furthermore, we provide some
asymptotic bounds. The problem coincides with the well known private
information retrieval problem based on a coded database to reduce the storage
overhead, when each linear combination contains exactly one information symbol.
  If any multiset of size $k$ of linear combinations from the linearly
independent information symbols can be recovered by $k$ disjoint subset of
servers, then the servers form a functional $k$-batch code. A~functional
$k$-batch code is a functional $k$-PIR code, where all the $k$ linear
combinations in the multiset are equal. We provide some bounds on the number of
servers for functional $k$-batch codes. In particular we present a random
construction and a construction based on simplex codes, WOM codes, and RIO
codes.
