MSR Codes with Linear Field Size and Smallest Sub-packetization for Any Number of Helper Nodes

The sub-packetization
$\ell$
and the field size
$q$
are of paramount importance in MSR code constructions. For optimal-access MSR codes, Balaji et al. proved that
$\ell\geq s^{\lceil n/s\rceil}$
, where
$s= d-k+1$
. Rawat et al. showed that this lower bound is attainable for all admissible values of
$d$
when the field size is exponential in
$n$
. After that, tremendous efforts have been devoted to reducing the field size. However, so far, reduction to a linear field size is only available for
$d\in\{k+1, k+2, k+3\}$
and
$d=n-1$
. In this paper, we construct the first class of explicit ontimal-access MSR codes with the smallest sub-packetization
$\ell=s^{\lceil n/s\rceil}$
for all
$d$
between
$k+1$
and
$n-1$
, resolving an open problem in the survey (Ramkumar et al., Foundations and Trends in Communications and Information Theory: Vol. 19: No. 4). We further propose another class of explicit MSR code constructions (not optimal-access) with an even smaller sub-packetization
$s^{\lceil n/(s+1)\rceil}$
for all admissible values of
$d$
, making significant progress on another open problem in the survey. Previously, MSR codes with
$\ell= s^{\lceil n/(s+1)\rceil}$
and
$q=O(n)$
were only known for
$d=k+1$
and
$d=n-1$
. The key insight that enables a linear field size in our construction is to reduce
$\binom{n}{r}$
global constraints of non-vanishing determinants to
$O_{s}(n)$
local ones, which is achieved by carefully designing a group of kernel matrices and then blowing them up to get parity check submatrices.