Algebraic lattices via polynomial rings
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(2019) 38:163
Algebraic lattices via polynomial rings Agnaldo José Ferrari1 · Antonio Aparecido de Andrade2 Received: 13 March 2018 / Revised: 4 August 2019 / Accepted: 24 September 2019 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019
Abstract Signal constellations having lattice structure have been studied as meaningful means for signal transmission over Gaussian channel. Usually the problem of finding good signal constellations for a Gaussian channel is associated with the search for lattices with high packing density, where in general the packing density is usually hard to estimate. The aim of this paper was to illustrate the fact that the polynomial ring Z[x] can produce lattices with maximum achievable center density, where Z is the ring of rational integers. Essentially, the method consists of constructing a generator matrix from a quotient ring of Z[x]. Keywords Galois ring · Lattice · Packing density · Center density Mathematics Subject Classification 11H06 · 11R80 · 97N70
1 Introduction The sphere packing problem is to find out how densely identical spheres can be packed together in the Euclidean space (Conway and Sloane 1998; Campello and Strapasson 2013). The packing density, (), of a lattice is the proportion of the space Rn covered by the nonoverlapping spheres of maximum radius centered at the points of . Due to the homogeneity n of the spheres’ distribution in Rn , the packing density is given by () = ρ vol(B(1)) 1 , where det() 2
ρ is the packing radius of the lattice and vol(B(1)) is the volume of the unit n-dimensional n sphere. The center density of a packing is defined by δ() = ρ 1 . The densest possible det() 2
lattice packings have only been determined in dimensions 1–8 and 24 (Conway and Sloane 1998; Cohn and Kumar 2009). It is also known that these densest lattice packings are unique.
Communicated by Thomas Aaron Gulliver.
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Antonio Aparecido de Andrade [email protected] Agnaldo José Ferrari [email protected]
1
School of Sciences, São Paulo State University (Unesp), Bauru, SP, Brazil
2
Department of Mathematics, São Paulo State University (Unesp), São José do Rio Preto, SP, Brazil
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A. J. Ferrari, A. A. de Andrade
Signal constellations having lattice structure are commonly accepted as good means for transmission with high spectral efficiency. The problem of finding good signal constellations for the Gaussian channel can be restated in terms of lattice sphere packings. Good lattice constellations for the Gaussian channel can be carved from lattices with high sphere packing density. The linear and highly symmetrical structure of lattices usually simplifies the decoding task. In Boutros et al. (1996), Boutros et al. present particular versions of the best lattice packings D4 , E 6 , E 8 , K 12 , 16 and 24 from totally complex algebraic cyclotomic fields. Andrade et al. (2010) proposed a technique for constructions of algebraic lattices in Euclidean space with optimal center density in dimensions 2, 3, 4,
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