Dense and decodable lattices in high dimensions We are currently looking for both dense and decodable lattices.
Precisely, we want a lattice which CVP can be solved in polynomial time like $O(n^2)$ or $O(n^3)$ where $n$ is the dimension like 128 or 512.
We have investigated E8 and Leech lattice. Their CVP can be solved efficiently but the dimensions are not high enough.
We also read the SPLAG (sphere packings lattices and groups). We notice they seldom talk about decoding algorithms.
Our current solution is to concatenate lots of Leech lattice.
 A: There are a number of sources of high-dimensional lattices that admit efficient CVP algorithms (and a few that admit efficient BDD algorithms, a "promise" version of CVP where one decodes errors contained in $\mathcal{B}_n(\lambda_1(L) / 2)$, e.g. the largest circumscribed sphere within the Voronoi cell of the lattice).
I'll briefly summarize some below, before discussing density.
CVP:

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*If one can efficiently solve CVP on $L$, then one can efficiently solve it on $L\otimes \mathbb{Z}^n \cong \bigoplus_{i = 1}^n L$. It sounds like you have already realized this, but it is worth mentioning explicitly.


*The lattices $A_n, A_n^*, D_n, D_n^*$ can all be efficiently decoded (this is mentioned in Conway and Sloane)


*A mild generalization of the $A_n, A_n^*$ lattices known as Coxeter lattices admit $O(n)$ CVP algorithms.
BDD:

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*Barnes-Wall Lattices admit $O(n(\log n)^2)$-time decoding algorithms


*A particularly dense family of lattices are the Discrete Logarithm lattices. The precise complexity of BDD on it depends on a number of choices one makes in constructing it, but looks like it can be $o(n^2)$ pretty easily.
There are other dense high-dimensional lattices that admit fairly efficient BDD algorithms (namely Barnes-Sloane lattices, which one can even list decode), but I think these are $\Omega(n^2)$ (but poly-time), although the algorithms are fairly naive, so with further work it is plausible they can be decoded in $o(n^2)$ time.
Of all of these, I believe the densest are the Barnes-Sloane lattices and the Discrete Logarithm lattices, so if you can tolerate a BDD algorithm instead of a CVP algorithm, I would look into the discrete logarithm lattices. If you require CVP algorithms, you should probably look into Coxeter lattices, although I do not know how their density will compare with using direct sums of the leech lattice.
