Sparse and Balanced MDS Codes Over Small Fields

Maximum Distance Separable (MDS) codes with a sparse and balanced generator matrix are appealing in distributed storage systems for balancing and minimizing the computational load. Such codes have been constructed via Reed-Solomon codes over large fields. In this paper, we focus on small fields. We prove that there exists an
$[n,k]_{q}$
MDS code that has a sparse and balanced generator matrix for any
$q\geq n-1$
provided that
$n\leq 2k$
, by designing several algorithms with complexity running in polynomial time in
$k$
and
$n$
. For the case
$n>2k$
, we give some constructions for
$q=n=p^{s}$
and
$k=p^{e}m$
based on sumsets, when
$e\leq s-2$
and
$m\leq p-1$
, or
$e=s-1$
and
$m < \frac {p}{2}$
.