we start from a suitable reduction so that the base (fine) lattice $\Lambda$ is not so thick (in terms of its *covering density*) and such that the kernels are not so sparse (in terms of *packing density*), we can bound the required size of $p$ to be within a finite range. For instance, we show that by starting from the family of Craig’s lattices [@Splag], we can build dense lattices from codes with alphabet-size $p = O((n\sqrt{\log n})^{1+\nu})$, where $\nu > 0$ is any (small) positive constant. This improves significantly the size of codes required by usual constructions [@Rush1989] [@Loeliger] (where $p$ cannot be smaller than $p = \omega(n^{3/2})$). As observed in [@Splag pp. 18-19], the works of Rush (and Loeliger) already significantly reduce the family sizes of typical proofs of the Minkowski-Hlawka lower bound. It is worth mentioning that, in terms of absolute family size, the best result is achieved by [@Zemor] by restricting the average to double-circulant codes or [@Moustrou] using cyclotomic lattices. For instance, while the logarithm of the search space for Craig’s lattices reductions has size $n^2 \log \log n$, the family based on double-circulant codes has size $n \log n$ (we refer the reader to 