The $(m,n,p)$-matrix multiplication tensor is a representation of the bilinear map $T\colon\mathbb{R}^{m\times n}\times\mathbb{R}^{n\times p}\rightarrow\mathbb{R}^{m\times p}$ given by $T(A,B)=AB$. Any low-rank decomposition of this tensor leads to a faster-than-naive matrix multiplication algorithm (à la the Strassen algorithm). Actually, to get a fast algorithm, it suffices for arbitrarily small perturbations of this tensor to have low rank; the minimum of such ranks is called the border rank.

**Where can I find a recent table of the best known upper and lower bounds on the rank and border rank of the $(m,n,p)$-matrix multiplication tensor for moderately small values of $m$, $n$ and $p$?**

Slide 24 here is close to what I want, though I would prefer explicit citations and larger $m$, $n$ and $p$.

Here's a related question that focuses on the $(3,3,3)$ case.

Tensors: geometry and applicationsby Landsberg? $\endgroup$ – Suvrit Sep 9 '16 at 15:28