On sequences of projections of the cubic lattice
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On sequences of projections of the cubic lattice Antonio Campello · João Strapasson
Received: 11 January 2012 / Revised: 17 August 2012 / Accepted: 25 November 2012 / Published online: 27 March 2013 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2013
Abstract In this paper we study sequences of lattices which are, up to similarity, projections of Zn+1 onto hyperplanes ν ⊥ for ν ∈ Zn+1 . We show a sufficient condition to construct sequences converging at rate O(1/ ν2/n ) to integer lattices and exhibit explicit constructions for some important families of lattices. The problem addressed here arises from a question of communication theory. Keywords
Projections · Lattices · Dense Packings
Mathematics Subject Classification (2010)
11H31 · 11H06 · 94A15
1 Introduction It was recently proved by Sloane et al. (2011) 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 Zn+1 onto a hyperplane determined by a linear equation with integer coefficients. Given a target lattice ⊂ Rn , it is possible to find a vector ν ∈ Zn+1 from the construction in Sloane et al. (2011), such that the distance between and a lattice
Communicated by José Eduardo Souza de Cursi. This work was supported by São Paulo Research Foundation (FAPESP) under grants 2009/18337-6 and 2011/01096-6. A. Campello (B) Department of Applied Mathematics, IMECC, University of Campinas, Campinas, SP 13083-859, Brazil e-mail: [email protected] J. Strapasson School of Applied Science, FCA, University of Campinas, Limeira, SP 13484-350, Brazil e-mail: [email protected]
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which is equivalent to the projection of Zn+1 onto ν ⊥ has order O(1/ ν1/n ), where ν is the Euclidean norm of ν. 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/ ν2/n ). We also show explicit constructions of such sequences for some families of lattices (Dn , odd n, Dn∗ ) and exhibit a table of which is, to our knowledge, the best sequences of projection lattices in the sense of the trade-off between density and ν. 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 (Vaishampayan and Costa 2003). 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 Zn+1 onto the subspace ν ⊥ for ν ∈ Zn+1 . Given a value l0 > 0, a worth objective to the design of good codes in the sense of Vaishampayan and Costa (2003) is th
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