I'm looking for references pertaining to the remark at the bottom of p.18 of Conway-Sloane, "Sphere Packings, Lattices and Groups" (3rd ed), henceforth referred to as "SPLG".

First, given a lattice Lambda in R^n, we define the density of a lattice packing of Lambda to be (cf SPLG p.10): $\Delta = V_n \rho ^n / (\det \Lambda)^{1/2}$

where

$V_n$ = volume of the unit sphere in $R^n$

$\rho$ = half the minimal distance between lattice points ("packing radius of $\Lambda$")

Minkowski (SPLG p.14) gave a non-constructive proof that there exists a lattice in dim n with $\Delta \geq \zeta(n) / 2^{n-1}.$

Finally, at the bottom of p.18 of SPLG the authors notes that

"... there are algorithms that would theoretically enable on to write down generating vectors of such a lattice in bound time."

Can anyone provide references of such algorithms? THANKS!

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