On rotated D n -lattices constructed via totally real number fields

Algebraic constructions of densest lattices

“…A broader question to be investigated is if algebraic constructions of lattices, as the ones approached here, can provide the greatest possible relative minimum product distance for rotated densest lattices in other dimensions. Table 1 Relative minimum product distance versus center density (from [6,21,17] and the results presented here). …”

“…If it were possible to construct such rotated D 3 and D 5 -lattices via principal ideals of Z[ζ 7 + ζ − 1 7 ] and Z[ζ 11 +ζ − 1 11 ], respectively, their minimum product distances would be twice those obtained in such constructions via Z-modules. However, in [18,Proposition 2.7] it was shown that if K is a totally real Galois extension with d K an odd integer, then it is impossible to construct rotated D n -lattices via fractional ideals of O K . In particular, it is impossible to construct rotated D 3 , D 4 and D 5 -lattices via fractional ideals of any Galois extension K ⊆ Q(ζ m ) with m odd.…”

“…When c = 1 the similar lattices Λ 1 and Λ 2 are said to be congruent or isomorphic. In this paper, as in [6,17], we will say that Λ 1 is a rotated Λ 2 -lattice if Λ 1 and Λ 2 are congruent.…”

“…In what follows we present a necessary condition for constructing an algebraic lattice Λ = σ α (I) with determinant D via a fractional ideal I of O K . Proposition 3.9 is a generalization of [18,Proposition 4.3], which considers rotated D n -lattices. …”

“…Special algebraic lattice constructions can be used to obtain certain lattice parameters such as packing density and minimum product distance, which are usually difficult to calculate for general lattices in R n . Some constructions and properties of algebraic lattices can be found in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. We quote particularly the paper [8], where full diversity rotated versions of the lattices A 2 , E 6 , E 8 , K 12 and Λ 24 are constructed.…”