  A \textit{functional $k$-batch} code of dimension $s$ consists of $n$ servers
storing linear combinations of $s$ linearly independent information bits. Any
multiset request of size $k$ of linear combinations (or requests) of the
information bits can be recovered by $k$ disjoint subsets of the servers. The
goal under this paradigm is to find the minimum number of servers for given
values of $s$ and $k$. A recent conjecture states that for any $k=2^{s-1}$
requests the optimal solution requires $2^s-1$ servers. This conjecture is
verified for $s\leq 5$ but previous work could only show that codes with
$n=2^s-1$ servers can support a solution for $k=2^{s-2} + 2^{s-4} +
\left\lfloor \frac{ 2^{s/2}}{\sqrt{24}} \right\rfloor$ requests. This paper
reduces this gap and shows the existence of codes for $k=\lfloor
\frac{2}{3}2^{s-1} \rfloor$ requests with $n=2^s-1$ servers. Another
construction in the paper provides a code with $n=2^{s+1}-2$ servers and
$k=2^{s}$ requests, which is an optimal result. %We provide some bounds on the
minimum number of servers for functional $k$-batch codes. These constructions
are mainly based on Hadamard codes and equivalently provide constructions for
\textit{parallel Random I/O (RIO)} codes.
