In this paper we study sequences of lattices which are, up to similarity, projections of $\mathbb{Z}^{n+1}$ onto a hyperplane $\bm{v}^{\perp}$, with $\bm{v} \in \mathbb{Z}^{n+1}$ and converge to a target lattice $\Lambda$ which is equivalent to an integer lattice. We show a sufficient condition to construct sequences converging at rate $O(1/ |\bm{v}|^{2/n})$ and exhibit explicit constructions for some important families of lattices.
It was recently proved [3] that any n-dimensional lattice can be approximated by a sequence of lattices such that each element is, up to similarity, the orthogonal projection of the cubic lattice Z n+1 onto a hyperplane determined by a linear equation with integer coefficients. Given a target lattice Λ ⊂ R n , it is possible to find a vector v ∈ Z n+1 from the construction in [3], such that the distance between Λ and a lattice which is equivalent to the projection of Z n+1 onto v ⊥ has order O(1/ v 1/n ), where v is the Euclidean norm of v. A natural question that arises from that result is whether it is possible to improve this convergence. We give a positive answer to this question by showing a sufficient condition to obtain sequences converging to an integer lattice with order O(1/ v 2/n ).
We also show explicit constructions of such sequences for some families of lattices (D n , odd n, D * n ) and exhibit a table of which is, to our knowledge, the best sequences of projection lattices in the sense of the tradeoff between density and v .
Apart from the purely geometric interest, the problem of finding sequences of projection lattices with a better order of convergence is motivated by an application in joint source-channel coding of a Gaussian channel [6]. In the aforementioned paper, the authors propose a coding scheme based on curves on flat tori and show that the efficiency of this scheme is closely related to the “small-ball radius” of these curves, which can be approximated by the packing radius of a lattice obtained by projecting Z n+1 onto the subspace v ⊥ for v ∈ Z n+1 . Given a value l 0 > 0, a worth objective to the design of good codes in the sense of [6] is the one of choosing a vector v ∈ Z n+1 with v = l 0 in such a way to maximize
which is the length of the shortest vector of Λ v , the projection of Z n+1 onto v ⊥ . Let δ Λv be the center density of these lattices (for undefined terms see Section II). Since the volume of Λ v is given by 1/ v (see [4]), we have:
therefore maximizing r(v) implies maximizing δ Λv .
Another geometrical formulation to this problem is the so-called fat strut problem. A “strut” is defined as a cylinder anchored at two points in Z n+1 such that its interior does not contain any other integer point. Given l 0 > 0, the fat strut problem asks for a vector v ∈ Z n+1 of length l 0 that maximizes the radius of the strut anchored at 0 and v. This problem is shown to be equivalent to the one of finding dense projections of Z n+1 [4]. Therefore, projection lattices with higher densities imply fat-struts with larger radii, and the problem addressed in this work is related to finding small vectors that attain high density projection lattices. This is done by considering families of projections of Z n+1 . This paper is organized as follows. In Section 2 we summarize some relevant concepts and results on lattices. In Section 3, we derive a sufficient condition to construct good sequences of projection lattices and in Section 4 we exhibit explicit constructions for some well-known lattices. Finally, in Section 5 we present our conclusions.
In this section we give a brief review of some relevant concepts concerning lattices and establish the notation to be used from now on. Given m linearly independent vectors b 1 , . . . , b m in R n , a lattice Λ is the set of all integer linear combinations of these vectors. The matrix G whose rows are the vectors b i is called a generator matrix for Λ and the matrix A = GG t is said to be a Gram matrix for Λ. The determinant or discriminant of Λ is defined as det Λ = det A and corresponds to the square of the volume of any fundamental region for the lattice Λ. We say that two lattices with generator matrices G 1 and G 2 are equivalent if there exists an unimodular matrix U , an orthogonal matrix Q and a real number c such that G 1 = c U G 2 Q. The density ∆ of a lattice is the ratio between the volume of a sphere of radius ρ (half of the minimal distance between two distinct lattices points) and the volume of a fundamental region, while the center density is defined as δ = ∆/V n where V n is the volume of the unitary sphere in R n . Sometimes we will refer to the center density of a specific lattice Λ as δ Λ .
Let G be a full-rank generator matrix for Λ. The dual lattice Λ * of Λ is the set of all x ∈ span(G) such that x, y is an integer number for all y ∈ Λ, where span(G) is the row space of G. One can easily verify that (GG t ) -1 G generates Λ * . We say that Λ is an integer (or rational ) lattice if its generator matrix has integer (rational) entries. All rational lattices are integers up to scale. The cubic lattice Z n is the full-dimensional integer self-dual lattice that has the canonical vectors e 1 = (1, 0, . . . , 0), . . . , e n = (0, . . . , 0, 1) as a basis. A list of the densest known packings in some dimensions as well as many other information about lattices can be found in [1].
We say that a sequence of lattices Λ w c
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