Almost Optimal Construction of Functional Batch Codes Using Extended Simplex Codes

A
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
$s \leqslant 5$
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
$k=\lfloor \frac {5}{6}2^{s-1} \rfloor - s$
requests with the same number of 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. These constructions are mainly based on extended Simplex codes and equivalently provide constructions for
parallel Random I/O (RIO)
codes.