  The MDS property (aka the $k$-out-of-$n$ property) requires that if a file is
split into several symbols and subsequently encoded into $n$ coded symbols,
each being stored in one storage node of a distributed storage system (DSS),
then an user can recover the file by accessing any $k$ nodes. We study the
so-called $p$-decodable $\mu$-secure erasure coding scheme $(1 \leq p \leq k -
\mu, 0 \leq \mu < k, p | (k-\mu))$, which satisfies the MDS property and the
following additional properties:
  (P1) strongly secure up to a threshold: an adversary which eavesdrops at most
$\mu$ storage nodes gains no information (in Shannon's sense) about the stored
file,
  (P2) partially decodable: a legitimate user can recover a subset of $p$ file
symbols by accessing some $\mu + p$ storage nodes.
  The scheme is perfectly $p$-decodable $\mu$-secure if it satisfies the
following additional property:
  (P3) weakly secure up to a threshold: an adversary which eavesdrops more than
$\mu$ but less than $\mu+p$ storage nodes cannot reconstruct any part of the
file.
  Most of the related work in the literature only focused on the case $p = k -
\mu$. In other words, no partial decodability is provided: an user cannot
retrieve any part of the file by accessing less than $k$ nodes.
  We provide an explicit construction of $p$-decodable $\mu$-secure coding
schemes over small fields for all $\mu$ and $p$. That construction also
produces perfectly $p$-decodable $\mu$-secure schemes over small fields when $p
= 1$ (for every $\mu$), and when $\mu = 0, 1$ (for every $p$). We establish
that perfect schemes exist over \emph{sufficiently large} fields for almost all
$\mu$ and $p$.
