The lifting construction: A general solution for the fat strut problem

Abstract

A cylinder anchored at two distinct points of the lattice Z n is called a strut if its interior does not contain a lattice point. We address the problem of constructing struts of maximal radius in Z n . Our main result is a general construction technique, which we call the lifting construction, which produces a sequence of struts that are optimal in the limit. We also tighten a previous result of ours-an achievable lower bound on the volume of a strut. The problem is motivated by a nonlinear analog communication problem. We demonstrate, through simulation, improvements in performance that are obtained using our construction.

Curves on Flat Tori and Analog Source-Channel Codes

“…, 1) to encode the information, then the scheme proposed here is exactly the one analysed in [11]. As we show next, for M > 1 the curves presented here outperform the ones presented in [11] (and also in [8]) in terms of length and small-ball radius.…”

“…In what follows we present two results: 1) A constructive scheme based on Example V-B that increases of N ! the lengths of the scheme proposed in [11], even with the improvements of the Lifting Construction [8]. This will yield an asymptotic behavior comparable to O(1/P N ), but with a better performance when N increases.…”

“…We adopt here the lattice terms as in [2]. For our purposes, the proximity measure for lattices will be the distance between their Gram matrices, as in [8]. This notion measures how close a lattice is to another one up to congruence transformations (rotations or reflections).…”

“…In Figure 4, the first curve from the bottom to the top represents the exponential sequence [11]. The second one is obtained by directly applying the Lifting Construction [8] to the hexagonal lattice and the last one displays the total length associated to our scheme.…”

Flat tori, lattices and spherical codes

“…We want to find a point a a a ∈ Z L such that the cylinder anchored at the origin and a a a does not contain any other lattice point and has maximal volume. The Lifting Construction [26] gives a general solution for this problem. It is shown in [27] how to construct a sequences of lattices which are, up to equivalence relations, similar to projections of Z L and arbitrarily close to any target (L − 1)-dimensional lattice.…”