Wang et al. (IEEE Transactions on Information Theory, vol. 62, no. 8, 2016) proposed an explicit construction of an $(n=k+2,k)$ Minimum Storage Regenerating (MSR) code with $2$ parity nodes and subpacketization $2^{k/3}$. The number of helper nodes for this code is $d=k+1=n-1$, and this code has the smallest subpacketization among all the existing explicit constructions of MSR codes with the same $n,k$ and $d$. In this paper, we present a new construction of MSR codes for a wider range of parameters. More precisely, we still fix $d=k+1$, but we allow the code length $n$ to be any integer satisfying $n\ge k+2$. The field size of our code is linear in $n$, and the subpacketization of our code is $2^{n/3}$. This value is slightly larger than the subpacketization of the construction by Wang et al. because their code construction only guarantees optimal repair for all the systematic nodes while our code construction guarantees optimal repair for all nodes.

翻译：Wang 等人（《IEEE信息论汇刊》，第62卷，第8期，2016年）提出了一种具有2个校验节点、子分组大小为 $2^{k/3}$ 的 $(n=k+2,k)$ 最小存储再生（MSR）码的显式构造。该码的辅助节点数为 $d=k+1=n-1$，在现有所有相同 $n,k$ 和 $d$ 的MSR码显式构造中，该码具有最小的子分组大小。本文针对更广泛的参数范围提出了一种新的MSR码构造方法。具体而言，我们仍固定 $d=k+1$，但允许码长 $n$ 取满足 $n\ge k+2$ 的任意整数。所构造码的域大小与 $n$ 呈线性关系，子分组大小为 $2^{n/3}$。该值略大于Wang等人构造的子分组大小，因为他们的码构造仅保证所有系统节点的最优修复，而我们的码构造保证所有节点的最优修复。