  The sub-packetization $\ell$ and the field size $q$ are of paramount
importance in the MSR array code constructions. For optimal-access MSR codes,
Balaji et al. proved that $\ell\geq s^{\left\lceil n/s \right\rceil}$, 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, till
now, reduction to 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
optimal-access MSR codes with the smallest sub-packetization $\ell =
s^{\left\lceil n/s \right\rceil}$ 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 even smaller sub-packetization $s^{\left\lceil n/(s+1)\right\rceil }$ for
all admissible values of $d$, making significant progress on another open
problem in the survey. Previously, MSR codes with $\ell=s^{\left\lceil
n/(s+1)\right\rceil }$ 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 the parity check
matrices.
