  A functional PIR array code is a coding scheme which encodes some $s$
information bits into a $t\times m$ array such that every linear combination of
the $s$ information bits has $k$ mutually disjoint recovering sets. Every
recovering set consists of some of the array's columns while it is allowed to
read at most $\ell$ encoded bits from every column in order to receive the
requested linear combination of the information bits. Functional batch array
codes impose a stronger property where every multiset request of $k$ linear
combinations has $k$ mutually disjoint recovering sets. Locality functional
array codes demand that the size of every recovering set is restrained to be at
most $r$. Given the values of $s, k, t, \ell,r$, the goal of this paper is to
study the optimal value of the number of columns $m$ such that these codes
exist. Several lower bounds are presented as well as explicit constructions for
several of these parameters.
