Constructions of algebraic lattices

Constructions of algebraic lattices

A.A. Andrade^I,*; A.J. Ferrari^II; C.W.O. Benedito^III; S.I.R. Costa^IV

^I,IIIDepartment of Mathematics, IBILCE, UNESP, 15054-000 São José do Rio Preto, SP, Brazil

IIDepartment of Applied Mathematics, IMECC, UNICAMP, 13083-859 Campinas, SP, Brazil

IVDepartment of Mathematics, IMECC, UNICAMP, 13083-859 Campinas, SP, Brazil E-mails: [e-mail] / [e-mail] / [e-mail] / suelifime.unicamp.br

ABSTRACT

In this work we present constructions of algebraic lattices in Euclidean space with optimal center density in dimensions 2, 3, 4, 6, 8 and 12, which are rotated versions of the lattices Λ_n, for n = 2,3,4,6,8 and K₁₂. These algebraic lattices are constructed through twisted canonical homomorphism via ideals of a ring of algebraic integers.

Mathematical subject classification: 18B35, 94A15, 20H10.

Key words: algebraic lattice, algebraic number field, center density, twisted canonical homomorphism.

1 Introduction

The classical sphere packing problem, still unsolved even today, is to find out how densely a large number of identical spheres can be packed together. To state this another way, consider a large empty region, such as an aircraft hangar, and ask what is the greatest number of ball bearings that can be packed into this region. If instead of ball bearings we try to pack identical wooden cubes the answer becomes easy. But spheres do not fit together so well as cubes, and there is always some wasted space in between. No matter how cleverly the ball bearings are arranged, about one quarter of the space will not be used.

Elementary number theory has been very useful to the development of error correcting codes in the early age of coding theory and the theory of Euclidian lattices became of great interest for the design of dense signal constellations well suited for transmission over AWGN channel. Furthermore, algebraic number theory has been very useful in mathematical tool that enables the design of good coding schemes for the fading channels (wireless communications). Algebraic lattices defined over algebraic number fields have been studied inseveral papers and from different points of view [1-7]. Giraud and Belfiore [8] proposed a technique for constructing signal sets suitable for the Rayleigh fading channel. The basic idea was to use lattice rotations to increase diversity, that is, the number of different values in the components of any two distinct points of the constellation. Boutros et al. [9] constructed rotated versions of lattices D₄, K₁₂ and Λ₆ via ideals of (ζ_n), for n = 8, 21 and 40, respectively.Bayer-Fluckiger [1-7] constructed rotated versions of lattices A_p-1, where p is an odd prime number, D₄, E₆, E₈, K₁₂, Λ₂₄ and Craig's lattices . Thus, having the construction of rotated lattices as our goal, in this work we present algebraic lattices in Euclidean space with optimal center density in dimensions 2, 3, 4, 6, 8 and 12 making use of the twisted canonical homomorphism.

This work is organized as follows. In Section 2, we present basic results ofalgebraic number fields and lattices. In Section 3, we present results about algebraic lattices making use of the twisted canonical homomorphism σ and examples of algebraic lattices with optimal center density in dimensions 2, 3, 4, 6, 8 and 12. Finally, in Section 4, we give ours conclusions.

2 Basic results

In this section, we present some results of algebraic number theory and lattices [10, 11] that are important to the development of the next section.

2.1 Algebraic number theory

Let be an algebraic number field, i.e., is an algebraic extension of of finite degree n, and therefore [:] = n.

Definition 2.1. An element β ∈ is called an algebraic integer if thereexists a monic polynomial non-zero f(x) with coefficients in such that f(β) = 0. The set = {β ∈ : β is an algebraic integer } is a ring called ring of algebraic integers in .

Let be the ring of algebraic integers of . It can be shown that has a basis {x₁,x₂,...,x_n} over called an integral basis of .

Definition 2.2. Let be a non-zero ideal of . The norm of the ideal is defined as the number of elements of the quotient ring /, i.e., () = |/|.

Definition 2.3. Let σ₁,σ₂,..., σ_n be the monomorphisms of in . The trace and the norm of an element β ∈ over are defined, respectively, by

The discriminant of over is defined by

where {x₁,x₂,...,x_n} is an integral basis of .

2.2 Packing lattice

In this section we present basic facts about packing lattices. The classical problem of the sphere packing consists in to find identical spheres in

ⁿ such that the proportion of the space that is occupied by the spheres is optimal.

Definition 2.4. An additive subgroup Λ ⊆ ⁿ is a lattice if there exists a basis B = {x₁,x₂,...,x_n} in ⁿ such that

The set B is called a -basis of the lattice Λ.

Let Λ ⊂

ⁿ be a lattice with -basis B = {x₁,x₂,...,x_n}.

Definition 2.5. The set

is called fundamental region of Λ with respect to basis B.

Definition 2.6. The volume of the lattice Λ is defined as the module of determinant of the matrix M =, where x_i= (x_i1,x_i2,...,x_in), for i = 1,2,...,n, and denoted by ol(Λ) = |det(M)|. The matrix M is called a generator matrix for the lattice Λ.

For obtain optimal packing lattices we need to find spheres whose centersare points of a lattice Λ and that the intersection of any two spheres is only a point. For the determination of the radius of these spheres, note that fixed k > 0, the intersection of the compact set {x ∈ ⁿ; |x| < k} with the lattice Λ is a finite set. Therefore the number t = min{|x|; x ∈ Λ, x ≠ 0} is well defined. Furthermore, ρ(Λ) = t/2 is the greatest radius such that it is possible to obtain a packing lattice. If B(ρ(Λ)) is the sphere with center in the origin and radius ρ(Λ) then the packing density of Λ is defined by

where δ(Λ) = ρ(Λ)ⁿ/ol(Λ) is called the center density of the lattice Λ.

3 Algebraic lattice

The connections between lattices and algebraic number fields have been studied by many authors from Minkowski onwards [1-7]. In this section, we define the twisted canonical homomorphism [3, 5], and we present some results about algebraic lattices. Furthermore, we present some examples of algebraic lattices in

ⁿ with optimal center density.

Let be an algebraic number field of degree n and be the ring of algebraic integers of . Let σ_j: → be the n distinct monomorphisms of . If σ_j() ⊆ , say that σ_j is real, case contrary, σ_j is called imaginary. If all themonomorphisms are reals, is called a totally real field and if all the monomorphisms are imaginary, is called a totally complex field. If φ: → is the complex conjugation then for all j = 1,2,...,n, it follows that φ º σ_j = σ_k, for some k = 1,2,...,n, and that σ_j = σ_k if and only if σ_j() ⊂ . Hence if r₁ is the number of indices such that σ_j() ⊂ , we can ordered the monomorphisms σ₁,σ₂,...,σ_n of such manner that σ₁,σ₂,...,σ_r1 are the real monomorphisms and that σ_r1+r₂+j = for j = 1,..., r₂. Hencen-r₁ is an even number and it can be write as r₁+2r₂ = n.

Definition 3.1 [3, 5]. The twisted canonical homomorphism σ: → ⁿ is defined by

where α, x ∈ , σ_i(α) ∈ and α_i = σ_i(α) > 0, for i = 1,2,...,r₁+r₂, and the notations (β) and (β) are the real and imaginary parts of the complex number β, respectively.

In the Definition 3.1 taking α = 1 we have that σ₁ is the canonical homomorphism (or Minkowski) [10]. By Bayer-Fluckiger [3], it follows that σ() is an algebraic lattice in ⁿ, where ⊆ is an ideal. By Samuel [10] it follows that the volume of σ() is given by

where x₁,x₂,...,x_n is a Z-basis of . Furthermore, if is a totally real field (or totally complex) then the volume of σ() is given by

Let be a totally real number field (or totally complex) of finite degree n, be the discriminant of and be the ring of algebraic integers in .

Proposition 3.2 [10]. If is a non-zero ideal of then the volume of σ() and the volume of σ() are given, respectively, by

Proof. We have that ol(σ()) = 2^-r2|(α), because = det(σ_i(x_k))², where {x₁,x₂,...,x_n} is a -basis of . For the second formula, since / is isomorph to σ_α()/σ_α() it follows that σ_α () is an additive subgroup of σ_α() of index (). Furthermore, since a fundamental domain of σ_α() is a disjoint union of () copies of a fundamental domain of σ_α() it follows that Vol(σ_α()) = Vol(σ_α())() and consequently Vol(σ_α()) = 2^-r2|(α)().

By Conway and Sloane [11] we have that if x ∈ then

where c_α = 1 if is a totally real field, c_α = if is a totally complex field and is the complex conjugate of the element x. Therefore ρ(σ_α()) = min{|σ_α()|: x ∈, x ≠ 0} = min{: x ∈, x ≠ 0}, whereis an ideal of .

Proposition 3.3. If

is a non-zero ideal ofthen thecenter density of the latticeσ_α() is given by

where t_α = min{: x ∈ , x ≠ 0}.

Proof. Let t_α = min{(αx), x ∈, x ≠ 0}. If is a totally real field then

and if is a totally complex field then

Therefore the center density is the same in both cases.

Example 3.4. If = (ζ₆), where ζ₆ is a primitive 6-th root of unity, α = 2 and = (1-ζ₆) is an ideal of = [ζ₆], then [: ] = 2, = -3, (α) = 4 and () = 1. If x ∈ then x = (a₀ + a₁ζ₆)(1-ζ₆), with a₀,a₁∈ , and thus (αx) = 4(+ +a₀a₁). Therefore, t_α = min{(αx): x ∈, x ≠ 0} = 4, with a₀ = 1 and a₁ = 0, and the center density of the lattice σ_α() is given by

which is the optimal center density for this dimension, i.e., with the same center density of the lattice Λ₂.

Similarly, in the next table, we have that the lattice σ_α(), where A is an ideal of = [ζ₆], has the same center density that the hexagonal lattice A₂Λ₂.

Example 3.5 [12]. The polynomial p(x) = x³-6x²+9x-1 is irreducible over . Since p(0) = -1, p(1) = 5, p(3) = -1 and p(4) = 3, it follows that the roots of p(x) are reals. Let x₁, x₂ and x₃ be the roots of p(x) and = (x₁). If Λ is a lattice with basis e₁ = (x₁,x₂,x₃), e₂ = (x₃,x₁,x₂) and e₃ = (x₂,x₃,x₁), then

is the generator matrix of the lattice Λ and det(M) = 54. Furthermore, if x ∈ Λ then x = a₁e₁+a₂e₂+a₃e₃, with a₁,a₂,a₃∈ . Thus |x|² = 18(+++a₁a₂+a₁a₃+a₂a₃), and therefore t = min{|x|; x ∈ Λ, x ≠ 0} =, with a₁ = 1 and a₂ = a₃ = 0. Hence the center density of the lattice Λ is given by

which is the optimal center density for this dimension, i.e., with the same center density of the lattice Λ₃. In general, if p(x) = x³+ax²+bx+c is irreducible over , where a,b,c ∈ , a² = 4b and c(27c+4a³-18ab) < 0, then the lattice Λ has the same center density that the lattice A₃

D₃Λ₃.

Example 3.6. If = (ζ₈), where ζ₈ is a primitive 8-th root of unity,= (ζ₈ +) is an ideal of = [ζ₈] and α = 3 -2(ζ₈ +) ∈ , then [:] = 4, = 256, () = 2 and (α) = 1. If x ∈then x = (ζ₈ + )(a₀ + a₁ζ₈ +a₂ + a₃), with a₀,a₁,a₂,a₃∈ , and thus

Hence t_α = min{(αx): x ∈ , x ≠ 0} = 8, with a₀ = 1 and a₁ = a₂ = a₃ = 0, and therefore the center density of the lattice σ_α() is given by

which is the optimal center density for this dimension, i.e., with the same center density of lattice Λ₄.

Similarly, in the next table, we have that the lattice σ_α(), where is an ideal of = [ζ₈], has the same center density that the lattice D₄Λ₄.

Example 3.7. If = (ζ₉), where ζ₉ is a primitive 9-th root of unity, = (1-ζ₉- ζ₉²- ζ₉⁴- ζ₉⁵) is an ideal of = [ζ₉] and α = 4 + 2ζ₉²+ 2ζ₉^-2∈ , then n = [:] = 6, = 3⁹, () = 9 and (α) = 64. If x ∈then x = (1- ζ₉- ζ₉²- ζ₉⁴- ζ₉⁵)(a₀+ a₁ζ₉+ a₂ζ₉²+ a₃ζ₉³+ a₄ζ₉⁴+ a₅ζ₉⁵), where a₀,...,a₅∈ , and thus (αx) = + 72a₀a₁ + + 36a₀a₂ + 72a₁a₂ + - 72a₀a₃ +36a₁a₃ + 72a₂a₃ +- 108a₀a₄ - 72a₁a₄ +36a₂a₄ + 72a₃a₄ +- 108a₀a₅ - 108a₁a₅ -72a₂a₅ + 36a₃a₅ + 72a₄a₅ +. Hence t_α = min{(αx): x ∈, x ≠ 0} = 36, with a₀ = a₄ = -1, a₁ = a₃ = a₅ = 0 and a₂ = 1, and therefore the center density of the lattice σ_α() is given by

which is the optimal center density for this dimension, i.e., with the same center density of the lattice Λ₆.

Similarly, in the next table, we have that the lattice σ_α(), whereis an ideal of = [ζ₉], has the same center density that the lattice E₆Λ₆.

Example 3.8. If = (ζ₂₀), where ζ₂₀ is a primitive 20-th root of unity,= (2+ 2ζ₂₀- + + - - ) is an ideal of = [ζ₂₀] and α = 5+5(ζ₂₀+ )+5+ )+3(+ ) ∈ , then n = [:] = 8, = 2⁸ 5⁶, () = 16 and (α) = 25. If x ∈ then x = (2+2ζ₂₀-++--)(a₀+a₁ζ₂₀+a₂+a₃+a₄+a₅+a₆+a₇), where a₀,...,a₇∈ , and thus (αx) = 1552 + 2944a₀a₁ + 1552 + 2496a₀a₂ +2944a₁a₂ + 1552 + 1808a₀a₃ + 2496a₁a₃ +2944a₂a₃ + 1552 +944a₀a₄ + 1808a₁a₄ + 2496a₂a₄+ 2944a₃a₄ + 1552+ 944a₁a₅ + 1808a₂a₅ +2496a₃a₅ + 2944a₄a₅ + 1552- 944a₀a₆ + 944a₂a₆ +1808a₃a₆ + 2496a₄a₆ + 2944a₅a₆ [phone]a₀a₇- 840a₁a₇ + 32a₂a₇ + 912a₃a₇ +1704a₄a₇ + 2336a₅a₇+ 2760a₆a₇ + 1360. Hence t_α = min{(αx) : x ∈, x ≠ 0} = 40, with a₀ = a₃ = 0, a₁ = a₅ = a₆ = -1 and a₂ = a₄ = a₇ = 1, and therefore the center density of the lattice σ_α() is given by

which is the optimal center density for this dimension, i.e., with the samecenter density of the lattice Λ₈.

Similarly, in the next table, we have that the lattice σ_α(), whereis an ideal of = [ζ₂₀], has the same center density that the lattice E₈Λ₈.

Example 3.9. If = (ζ₂₁), where ζ₂₁ is a primitive 21-th root of unity,= ( - + ) is an ideal of = [ζ₂₁] and α = 1, then n = [:] = 12, = 3⁶ 7¹⁰, () = 7. If x ∈ then x = ( - +)(a₀+ a₁ζ₂₁ + a₂ +a₃ + a₄ + a₅ +a₆ + a₇ + a₈ +a₉ + a₁₀+a₁₁), where a₀,...,a₁₁∈ , and thus (αx) = 28 +28 + 28a₀a₁₀ - 14a₁a₁₀ + 28+28a₀a₁₁ + 28a₁a₁₁ + 28- 14a₀a₂ -14a₁₁a₂ +28 - 14a₀a₃ - 14a₁a₃ - 28a₁₀a₃ +28 -14a₀a₄ - 14a₁a₄ - 28a₁₁a₄ - 14a₂a₄ +28 +28a₀a₅ - 14a₁a₅ + 28a₁₀a₅ - 14a₂a₅ -14a₃a₅ + 28 + 28a₁a₆ - 14a₁₀a₆ + 28a₁₁a₆ -14a₂a₆ - 14a₃a₆ - 14a₄a₆ + 28 - 28a₀a₇ -14a₁₀a₇ -14a₁₁a₇ + 28a₂a₇ - 14a₃a₇ - 14a₄a₇ -14a₅a₇ +28 - 28a₁a₈ - 14a₁₀a₈ - 14a₁₁a₈ +28a₃a₈ -14a₄a₈ - 14a₅a₈ - 14a₆a₈ + 28-14a₀a₉ -14a₁₁a₉ - 28a₂a₉ + 28a₄a₉ - 14a₅a₉ -14a₆a₉ -14a₇a₉ + 28. Hence t_α = min{(αx) : x ∈, x ≠ 0} = 28, with a₀ = a₁ = a₂ = a₃ = a₄ = a₅ = a₆ = a₈ = a₉ = a₁₀ = a₁₁ = 0 and a ₇ = -1, and therefore the center density of the lattice σ_α() is given by

which is the optimal center density for this dimension, i.e., with the same center density of the lattice K₁₂.

Similarly, in the next table, we have that the lattice σ_α(), whereis an ideal of = [ζ₂₁] and α = 1, has the same center density that the lattice K₁₂.

4 Conclusions

In this work we presented examples of algebraic lattices via the twisted canonical homomorphism with optimal center density in dimensions 2, 3, 4, 6, 8 and 12. These algebraic lattices are rotated versions of known dense lattices. Note that the examples given in this work are not new either. What is new however is the way the densities of the lattices are checked through computations rather than by theoretic arguments. Furthermore, with the use of canonical homomorphism we believe that it is possible to construct algebraic lattices with optimal center density in other dimensions.

Acknowledgements. The authors would like to thank the anonymous reviewers for their insightful comments that greatly improved the quality of this work.

Received: 17/IX/09.

Accepted: 17/V/10.

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*

Financial support by FAPESP 2007/56052-8, 2005/04177-6 and 2007/06381-5.

Publication Dates

• Publication in this collection
  22 Nov 2010
• Date of issue
  2010

History

• Received
  17 Sept 2009
• Accepted
  17 May 2010