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ℓ和域大小q在MSR阵列码构造中至关重要。对于最优访问MSR码，Balaji等人证明ℓ≥s^⌈n/s⌉，其中s=d-k+1。Rawat等人表明，当域大小在n上呈指数增长时，该下界对所有允许的d值均可达到。此后，大量研究致力于减小域大小。然而，迄今为止，线性域大小的实现仅适用于d∈{k+1,k+2,k+3}和d=n-1的情形。本文构造了首个显式最优访问MSR码类，其对k+1至n-1之间的所有d值均能达到最小子分组化ℓ = s^⌈n/s⌉，从而解决了综述（Ramkumar等，《通信与信息理论基础与趋势》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显式MSR码构造（非最优访问），其对所有允许的d值实现了更小的子分组化s^⌈n/(s+1)⌉，在解决该综述中另一个开放问题上取得重要进展。此前，仅d=k+1和d=n-1情形已知存在ℓ=s^⌈n/(s+1)⌉且q=O(n)的MSR码。实现线性域大小的关键洞察在于通过精心设计奇偶校验矩阵，将非零行列式的Ɑ(n,r)个全局约束简化为Os(n)个局部约束。