  An $(n, k, d, \alpha)$-MSR (minimum storage regeneration) code is a set of
$n$ nodes used to store a file. For a file of total size $k\alpha$, each node
stores $\alpha$ symbols, any $k$ nodes recover the file, and any $d$ nodes can
repair any other node via each sending out $\alpha/(d-k+1)$ 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---$\alpha$---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 $\alpha$.
Furthermore, it is less than other MSR codes' $\alpha$ for a subset of
practical parameters. We offer hints on how our code repairs multiple failures
at once.
