论文标题
多线性代数用于最低存储再生代码
Multilinear Algebra for Minimum Storage Regenerating Codes
论文作者
论文摘要
a $（n，k，d，α）$ - MSR（最小存储再生）代码是用于存储文件的$ N $节点的集合。对于总尺寸$kα$的文件，每个节点存储$α$符号，任何$ k $ nodes恢复文件，任何$ d $ nodes都可以通过每个发送$α/（d-k+1）$符号来修复任何其他节点。 在这项工作中，我们探索了使用偏斜的对称矩阵，多项式，对称代数和外部代数来重新表达臭名昭著的产品 - 矩阵结构的各种方法。然后，我们引入了一个多线性代数基金会来生产$ \ bigl（n，k，\ frac {（k-1）t} {t-1}，\ binom {k-1} {t-1} {t-1} \ bigr）$ - MSR代码，用于一般$ t \ geq2 $。在$ t = 2 $结束时，它们将产品垫构造作为特殊情况。在$ t = k $结束时，我们恢复模式的确定代码$ m = k $;进一步限制到$ n = k+1 $，使其与MSR点上的分层代码相同。我们的代码的子包装水平--- $α$ ---独立于$ n $和小。它小于$ l^{2.8（d-k+1）} $，其中$ l $是alrrabiah - guruswami的下限$α$。此外，对于实际参数的子集而言，它比其他MSR码的$α$要小。我们提供有关代码如何一次修复多个故障的提示。
An $(n, k, d, α)$-MSR (minimum storage regeneration) code is a set of $n$ nodes used to store a file. For a file of total size $kα$, each node stores $α$ symbols, any $k$ nodes recover the file, and any $d$ nodes can repair any other node via each sending out $α/(d-k+1)$ symbols. In this work, we explore various ways to re-express the infamous product-matrix construction using skew-symmetric matrices, polynomials, symmetric algebras, and exterior algebras. We then introduce a multilinear algebra foundation to produce $\bigl(n, k, \frac{(k-1)t}{t-1}, \binom{k-1}{t-1}\bigr)$-MSR codes for general $t\geq2$. At the $t=2$ end, they include the product-matrix construction as a special case. At the $t=k$ end, we recover determinant codes of mode $m=k$; further restriction to $n=k+1$ makes it identical to the layered code at the MSR point. Our codes' sub-packetization level---$α$---is independent of $n$ and small. It is less than $L^{2.8(d-k+1)}$, where $L$ is Alrabiah--Guruswami's lower bound on $α$. Furthermore, it is less than other MSR codes' $α$ for a subset of practical parameters. We offer hints on how our code repairs multiple failures at once.
