Small-d MSR Codes With Optimal Access, Optimal Sub-Packetization, and Linear Field Size

This paper presents an explicit construction of a class of optimal-access, minimum storage regenerating (MSR) codes, for small values of the number
$d$
of helper nodes. The construction is valid for any parameter set
$(n,k,d)$
with
$d \in \{k+1, k+2, k+3\}$
and employs a finite field
$\mathbb {F}_{q}$
of size
$q=O(n)$
. We will refer to the constructed codes as
$\text {Small-}\mathsf {d}$
MSR codes. The sub-packetization level
$\alpha $
is given by
$\alpha = s^{{\lceil \frac {n}{s}\rceil }}$
, where
$s=d-k+1$
. By an earlier result on the sub-packetization level for optimal-access MSR codes, this is the smallest value possible.