Print version ISSN 1677-1966On-line version ISSN 2179-8451

TEMA (São Carlos) vol.21 no.1 São Carlos Jan./Apr. 2020  Epub Apr 30, 2020

http://dx.doi.org/10.5540/tema.2020.021.01.0057

Articles

Constructions of Dense Lattices over Number Fields

1Departamento de Matemática, Instituto de Biociências, Letras e Ciências Exatas (Ibilce), Universidade Estadual Paulista “Júlio de Mesquita Filho” (Unesp), Campus de São José do Rio Preto - SP, Brasil. E-mail: antonio.andrade@unesp.br

2Departamento de Matemática, Faculdade de Ciências (FC), Universidade Estadual Paulista “Júlio de Mesquita Filho” (Unesp), Campus de Bauru - SP, Brasil. E-mail: agnaldo.ferrari@unesp.br

3Department of Mathematics & Statistics, San Diego State University, San Diego, California, USA. E-mail: interlan@sdu.edu

4Instituto Federal de São Paulo, Cubatão - SP, Brasil. E-mail: dearaujorobsonricardo@gmail.com

ABSTRACT

In this work, we present constructions of algebraic lattices in Euclidean space with optimal center density in dimensions 2;3;4;5;6;8 and 12, which are rotated versions of the lattices Λ n , for n=2,3,4,5,6,8 and K 12. These algebraic lattices are constructed through canonical homomorphism via ℤ-modules of the ring of algebraic integers of a number field.

Keywords: algebric lattices; number fields; sphere packings

RESUMO

Neste trabalho, apresentamos construçõoes de reticulados algébricos no espaço euclidiano com densidade central ótima nas dimensões 2, 3, 4, 5, 6, 8 e 12, que são versões rotacionadas dos reticulados Λ n , para n=2,3,4,5,6,8 e K 12, onde esses reticulados algébricos são construídos através do homomorfismo canônico via ℤ-módulos do anel de inteiros algébricos de um corpo de números.

Palavras-chave: reticulados algébricos; corpos de números; empacotamento esférico

1 INTRODUCTION

Algebraic number theory has recently raised a great interest for its new role in algebraic lattice theory and in code design for many different coding applications. Algebraic lattices have been useful in information theory and the question of finding algebraic lattices over number fields maximum center density. The problem of finding algebraic lattices with maximal minimum product distance has been studied in last years and this has motived special attention of many researchs in considering ideals of certain rings. The search for dense algebraic lattices in general dimensions has been encouraged in the last decades because they can be applied to Information Theory 1) - (4.

The classical sphere packing problem consists to find out how densely a large number of identical spheres can be packed together in the Euclidean space. The packing density, ∆(Λ), of a lattice Λ is the proportion of the space ℝ n covered by the non-overlapping spheres of maximum radius centered at the points of Λ. The densest possible lattice packings have only be determined in dimensions 1 to 8 and 24. It is also known that these densest lattice packings are unique (up to equivalences) 5.

This paper is organized as follows. In Section 2, notions and results from algebraic number theory that are used in the work are reviewed. In Section 3, rotated lattices are constructed from number fields in dimensions 2,3,4,5,6,8 and 12, which are rotated versions of the lattices Ln, for n=2,3,4,5,6,8 and K 12.

2 BACKGROUND OF NUMBER FIELDS

Let 𝕂 be a number field, i.e., 𝕂 is a finite extension of ℚ. By Primitive Element Theorem, there is an element θ𝕂 such that K=(θ)=i=0n-1ai , where θ is a root of a polynomial p(x) [x] of minimal degree n. A cyclotomic field is a number field such that K=(θ) , where θ is a primitive n-th root of unity. If θ1=θ,θ2,...,θn are the n distinct roots of p(x), then threre are exactly n distinct ℚ-embeddings σi:K such that σi(θ)=θi , for all i=1,2,,n . Furhtermore, there are r 1 real embeddings σ1,,σr1 and 2r 2 complex embeddings σr1+1,σr1+1,,σr1+r2,σr1+r2 . If ℜ(x) and ℑ(x) denote, respectively, the real part and the imaginary part of x, the canonical embedding σ:Kn , with xK , is defined by

σx=σ1x,,σr1x,Rσr1+1x,,Jσr1+r2x.

The set OK={α:f(α)=0 for some monic polynomial f(x)[x]} is a ring called ring of algebraic integers of 𝕂. The ring 𝒪 𝕂 has a basis α1,α2,,αn over ℤ. In other words, every element αOK is uniquely written as α=i=1naiαi , where αi for all i=1,2,,n , and every nonzero fractional ideal of 𝒪 𝕂 is a free ℤ-module of rank n7.

If αK , the value

TrKα=i=1nσiα

is called trace of α in 𝕂. If α1,α2,,αn is an integral basis of 𝕂, the discriminant of 𝕂 is defined as DK=detσjαi2 and it is an invariant over change of basis 6.

3 CONSTRUCTIONS OF DENSE ALGEBRAIC LATTICES

A lattice ∧ is a discrete additive subgroup of ℝ n , that is, 0n is a lattice iff there are linearly independent vectors v1,v2,...,vk , with kn , in ℝ n such that

Λ=i=1kaivi:ai, for all i=1,2,,k.

The set v1,v2,...,vk is called a basis for ∧, the matrix M whose rows are these vectors is called a generator matrix for ∧ and the matrix G=MMt is called Gram matrix.

If M is a ℤ-submodule in 𝕂 of rank n, the set Λ=σ(M) is a lattice in ℝ n called an algebraic lattice. The center density of Λ is given by

δΛ=tn/22nOK:MDK,

where t=min{TrK(αα):α,α0} and [𝒪K:] denotes the index of the submodule M.

Example 3.1. If K=(ζ3) , where ζ3 is the primitive 3-th root of unity, then [K:]=2,{1,ζ3} is a basis of 𝕂 and DK=-3 . If ℳ is a submodule of 𝒪 𝕂 given by

M=a0+a1ζ3:a0,a1,

then [𝒪K:]=1 and

TrKαα=2a02-a0a1+a12,

where α . Since t=min{TrK(aā):α;α0}=2 with a0=1 and a1=0 , it follows that

δM=2/223=123,

i.e., the center density of σ() is the same of the lattice4. Similarly, if K=3 , then [K:]=2,{1,3} is a basis of 𝕂 and DK=12 . If ℳ is a submodule of 𝒪 𝕂 given by =a0+a13:a0-a10(mod 2) and a0,a1 , then [𝒪K:]=2 and TrK(α2)=8a02+24a0a1+24a12 , where α . Since t=min{TrK(α2):α,α0}=8 with a0=1 and a1=0 , it follows that δ()=8/22233=143 .

Example 3.2. If K=(θ) , where θ=ζ9+ζ9-1 and ζ 9 is the primitive 9-th root of unity, then [K:]=3,{1,θ,θ2} is an integral basis of 𝕂 and DK=34 . If ℳ is a submodule of 𝒪 𝕂 given by

=a0+a1θ+a2θ2:a00(mod 2) and a0+2a1+a20(mod 3), where a0,a1,a2,

then [𝒪K:]=6 and

TrK(α2)=18(a02+a0a1+5a0a2+a12+5a1a2+9a22),

where α . Since t=min{TrK(α2):α,α0}=18 with a0=1 and a1=a2=0 , it follows that

δM=18/2354=142,

i.e., the center density of σ() is the same of the lattice Λ3 . Similarly, if =a0+a1θ+a2θ2:a00(mod 2) and a0+2a1+a20(mod 3), where a0,a1,a2 , then [𝒪K:]=6 and TrK(α2)=18(3a02+3a0a1+10a0a2+a12+5a1a2+9a22) , where α . Since t=min{TrK(α2):α,α0}=18 with a0=a2=0 and a1=1 , it follows that δ()=18/232·3·32142 . Similarly, if =a0+a1θ+a2θ2:a00(mod 2) and a0+2a1+a20(mod 3), where a0,a1,a2 , then [𝒪K:]=6 and TrK(α2)=18(3a02+3a0a1+10a0a2+a12+5a1a2+9a22) , where α . Since t=min{TrK(α2):α,α0}=18 with a0=a2=0 and a1=1 , it follows that δ()=18/232·3·32=142 . Finally, if K=(θ) , where θ is a root of p(x)=x33x+1 , then [K:]=3,{1,θ,θ2} is a basis of 𝕂 and DK=34 . If ℳ is a submodule of 𝒪 𝕂 given by =a0+a1θ+a2θ2:a20(mod 2) and a0a1+a20(mod 3), with a0,a1,a2 , then [𝒪K:]=6 and TrK(α2)=18(a02+5a0a1+3a0a2+9a12+10a1a2+3a22) , where α . Since t=min{TrK(α2):α,α0}=18 with a0=1 and a1=a2=0 , it follows that δ()=98/232·73=142 .

Example 3.3. If K=(ζ8) , where ζ 8 is the primitive 8-th root of unity, then [K:]=4,{1, ζ8,ζ82,ζ83} is an integral basis of 𝕂 and DK=28 . If ℳ is a submodule of 𝒪 𝕂 given by

M=a0+a1ζ8+a2ζ82+a383:a0+a1+a2+a30mod 2, where a0,a1,a2,a3,

then [OK:M]=2 and

TrKαα=82a02-2a0a3+a12-a1a2+a22-2a2a3+2a32,

where αM . Since t=minTrKα2:αM,α0=8 with a1=1 and a0=a2=a3=0 , it follows that

δM=8/2432=18,

i.e., the center density of σ() is the same of the lattice Λ4. Similarly, if M=a0+a1ζ8+a2ζ82+a3ζ83:a2+a30mod 2, where a0,a1,a2,a3 , then OK:M=2 and TrKα2=8a02+a12+a22+2a32+a0a2+2a0a3+a1a2+2a2a3 , where αM . Since t=minTrKα2:αM,α0=8 with a0=1 and a1=a2=a3=0 , it follows that δM=8/2432=18 . Similarly, if K=(θ) , where θ is a root of px=x4+3x2+1 , then K:=4 , where 1,θ,θ2,θ3 is an integral basis of 𝕂, DK=245˙2 . If ℳ is a submodule of 𝒪 𝕂 given by M=a0+a1θ+a2θ2+a3θ3:a0-2a1+2a2-a30mod 5, where a0,a1,a2,a3 , then OK:M=10 , and if αM , then TrKαα=40a02-40a0a1+132a0a2+360a0a3+20a12-28a1a2-140a1a3+158a22+720a2a3+900a32 . Since t=minTrKα2:αM,α0=20 with a0=a2=a3=0 and a1=1 , if follows that δM=20/2423·52=18 .

Example 3.4. If K=(θ) , where θ=ζ4410-ζ4412 and ζ 44 is the primitive 44-th root of unity, then [K:]=5,{1,θ,θ2,θ3,θ4} is an integral basis of 𝕂 and the discriminant of 𝕂 is 114 . Let ℳ be a submodule of 𝒪 𝕂 given by

={a0+a1θ+a2θ2+a3θ3+a4θ4:a00(mod 11),5a2+a30(mod 11)and a0+15a1+11a2+a40(mod 22), where a0,a1,a2,a3,a4}..

In this case, ℳ is a submodule of 𝒪 𝕂 of index 2·113 and the trace form of α is given by

TrK/α2=37752a02+43802a0a1+79134a0a2+16456a0a3+136488a0a4 +12826a12+46706a1a2+10406a1a3+79860a1a4+44286a22 +26136a2a3+144716a2a4+9438a32+30976a3a4+124388a42.

Thus, t=min{TrK(α2):α,α0}=242 with a0=a2=a3=0,a1=3 and a4=1 . As the volume of the lattice σ (mathcalM) is DKM:OK=2·115 , it follows that

δM=242/252·115=182,

i.e., the center density of σ() is the same of the lattice Λ7 .

Example 3.5. If K=(ζ9) , where ζ 9 is the primitive 9-th root of unity, then [K:]=6,{1,ζ9,ζ92,ζ93,ζ94,ζ95} is an integral basis of 𝕂 and DK=39 . If ℳ is a submodule of 𝒪 𝕂 given by

M={a0+a1ζ9+a2ζ92+a3ζ93+a4ζ94+a5ζ95:a1a2+a4a50(mod 3),where a0,a1,...,a5},,

then [𝒪K:]=9 and

TrK(αα)=18(a02+a0a1+a0a2+a0a3+2a0a4+2a0a5+a12+a1a3+3a1a4 +a22+a2a3+3a2a5+a32+2a3a4+2a3a5+3a42+3a52,

where α . Since t=min{TrK(αα):α,α0}=18 with a0=1 and a1=a2=a3=a4=a5=0 , it follows that

δM=18/26363=183,

i.e., the center density of σ() is the same of the lattice Λ6 .

Example 3.6. If K=(ζ20) , where ζ 20 is the primitive 20-th root of unity then [K:] =8,1,ζ20,ζ202,ζ203,ζ204,ζ205,ζ206,ζ207 is an integral basis fo 𝕂 and DK=28·56 . If ℳ is a submodule of 𝒪 𝕂 given by

M={a0+a1ζ20+a2ζ202+a3ζ203+a4ζ204+a5ζ205+a6ζ206+a7ζ207:a0+a40(mod 4), a1+a50(mod 2),a2+a3+a60(mod 4) and a70(mod 5), where a0,a1,...,a7},

then [𝒪K:]=5 and

TrK(αα)=20(2a02+2a0a1+5a0a2+3a0a3+3a0a4+2a0a5+5a0a6+8a0a7+a12 +3a1a2+2a1a3+2a1a4+a1a5+3a1a6+5a1a7+4a22+4a2a3+5a2a4 +3a2a5+7a2a6+12a2a7+2a32+3a3a4+2a3a5+5a3a6+7a3a7+2a42 +2a4a5+5a4a6+8a4a7+a52+3a5a6+5a5a7+4a62+12a6a7+10a72),

where α . Since t=min{TrK(αα):α,α0}=20 with a1=1 and a0=a2=a3=a4=a5=a6=a7=0 , it follows that

δM=20/2824·54=116,

i.e., the center density of σ() is the same of the lattice Λ8 .

Example 3.7. If K=(ζ21) , where ζ 21 is the primitive 21-th root of unity, then [K:]=12,1,ζ21,,ζ2111 is an integral basis of 𝕂 and DK=36.710 . If ℳ is a submodule of 𝒪 𝕂 given by

M=ζ216-ζ212+1a0+ζ217-ζ213+ζ21a1+ζ218-ζ214+ζ212a2 +ζ219-ζ215+ζ213a3+ζ2110-ζ216+ζ214a4+ζ2111-ζ217+ζ215a5 +ζ2111-ζ219+ζ214-ζ213+ζ21-1a6 +ζ2111-ζ2110-ζ219+ζ218-ζ216+ζ215-ζ213+ζ212-1a7 +-ζ2110+ζ218-ζ217-1a8+-ζ2111+ζ219-ζ218-ζ21a9 +-ζ2111+ζ2110-ζ218+ζ216-ζ214+ζ213-ζ212-ζ21+1a10 +-ζ218+ζ217+ζ216-ζ215-ζ212+1a11, where a0,a1,,a11},

then [𝒪K:]=7 and

TrKαα=28a02-14a0a2-14a0a3-14a0a4+28a0a5-28a0a7-14a0a9 +28a0a10+28a0a11+28a12-14a1a3-14a1a4-14a1a5+28a1a6 -28a1a8-14a1a10+28a1a11+28a22-14a2a4-14a2a5-14a2a6 +28a2a7-28a2a9-14a2a11+28a32-14a3a5-14a3a6-14a3a7 +28a3a8-28a3a10+28a42-14a4a6-14a4a7-14a4a8+28a4a9 -28a4a11+28a52-14a5a7-14a5a8-14a5a9+28a5a10+28a62 -14a6a8-14a6a9-14a6a10+28a6a11+28a72-14a7a9-14a7a10 -14a7a11+28a82-14a8a10-14a8a11+28a92-14a9a11+28a102 +28a112,

where α . Since t=min{TrK=(αα):α;α0}=28 with a0=1 and a1=a2==a11=0 , it follows that

δM=28/21233·56=133,

i.e., the center density of σ() is the same of the lattice K 12 .

ACKNOWLEDGMENT

The authors thank the reviewer for carefully reading the manuscript and for all the suggestions that improved the presentation of this work. The authors also thank FAPESP 2013/25977-7 and CNPq 429346/2018-2 for its financial support.

REFERENCES

1. A.A. Andrade & R. Palazzo Jr. Linear codes over finite rings. TEMA - Trends in Applied and Computational Mathematics, 6(2) (2005), 207-217. [ Links ]

2. A.S. Ansari, R. Shah, Zia Ur-Rahman & A.A. Andrade. Sequences of primitive and non-primitive BCH codes. TEMA - Trends in Applied and Computational Mathematics, 19(2) (2018), 369-389. [ Links ]

3. A. A. Andrade, A. J. Ferrari & C. W. O. Benedito, Constructions of algebraic lattices, Comput. Appl. Math., 29 (2010) 1-13. [ Links ]

4. E. Bayer-Fluckiger, Lattices and number fields, In: Contemp. Math., Amer. Math. Soc., Providence (1999), 69-84. [ Links ]

5. J. H. Conway & N. J. A. Sloane. Sphere Packings, Lattices and Groups, 3rd Edition, Springer Verlag, New York (1999). [ Links ]

6. J. C. Interlando, J. O. D. Lopes & T. P .N. Neto. The discriminant of abelian number fields, J. Algebra Appl., 5 (2006), 35-41. [ Links ]

7. P. Samuel. Algebraic Theory of Numbers, Hermann, Paris (1970). [ Links ]

Received: September 26, 2019; Accepted: January 01, 2020