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.

翻译：子分组化参数$\ell$和域大小$q$是MSR阵列码构造中至关重要的指标。对于最优访问MSR码，Balaji等人证明$\ell\geq s^{\left\lceil n/s \right\rceil}$，其中$s=d-k+1$。Rawat等人表明，当域大小关于$n$呈指数级增长时，该下界对所有允许的$d$值均可达到。此后，大量研究致力于降低域大小。然而迄今为止，仅当$d\in\{k+1,k+2,k+3\}$和$d=n-1$时才能实现线性域大小。本文针对所有介于$k+1$和$n-1$之间的$d$值，构造了首个具有最小子分组化$\ell = s^{\left\lceil n/s \right\rceil}$的显式最优访问MSR码，解决了综述文献（Ramkumar等，《通信与信息理论基础与前沿》，第19卷第4期）中的一个开放问题。我们进一步提出了另一类非最优访问的显式MSR码构造，对所有允许的$d$值实现了更小的子分组化$s^{\left\lceil n/(s+1)\right\rceil}$，这为综述中的另一个开放问题取得了重要进展。此前，仅当$d=k+1$和$d=n-1$时存在$\ell=s^{\left\lceil n/(s+1)\right\rceil}$且$q=O(n)$的MSR码。实现线性域大小的关键思路在于：通过精心设计校验矩阵，将$\binom{n}{r}$个非零行列式的全局约束缩减为$O_s(n)$个局部约束。