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 Λn , for 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 Λn , para 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 11. A.A. Andrade & R. Palazzo Jr. Linear codes over finite rings. TEMA - Trends in Applied and Computational Mathematics, 6(2) (2005), 207-217.) - (44. E. Bayer-Fluckiger, Lattices and number fields, In: Contemp. Math., Amer. Math. Soc., Providence (1999), 69-84..
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 ℝn 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) 55. J. H. Conway & N. J. A. Sloane. Sphere Packings, Lattices and Groups, 3rd Edition, Springer Verlag, New York (1999)..
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 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 such that , where θ is a root of a polynomial of minimal degree n. A cyclotomic field is a number field such that , where θ is a primitive n-th root of unity. If are the n distinct roots of p(x), then threre are exactly n distinct ℚ-embeddings such that , for all . Furhtermore, there are r 1 real embeddings and 2r 2 complex embeddings . If ℜ(x) and ℑ(x) denote, respectively, the real part and the imaginary part of x, the canonical embedding , with , is defined by
The set for some monic polynomial is a ring called ring of algebraic integers of 𝕂. The ring 𝒪 𝕂 has a basis over ℤ. In other words, every element is uniquely written as , where for all , and every nonzero fractional ideal ℳ of 𝒪 𝕂 is a free ℤ-module of rank n77. P. Samuel. Algebraic Theory of Numbers, Hermann, Paris (1970)..
If , the value
is called trace of α in 𝕂. If is an integral basis of 𝕂, the discriminant of 𝕂 is defined as and it is an invariant over change of basis 66. J. C. Interlando, J. O. D. Lopes & T. P .N. Neto. The discriminant of abelian number fields, J. Algebra Appl., 5 (2006), 35-41..
3 CONSTRUCTIONS OF DENSE ALGEBRAIC LATTICES
A lattice ∧ is a discrete additive subgroup of ℝn , that is, is a lattice iff there are linearly independent vectors , with , in ℝn such that
The set is called a basis for ∧, the matrix M whose rows are these vectors is called a generator matrix for ∧ and the matrix is called Gram matrix.
If M is a ℤ-submodule in 𝕂 of rank n, the set is a lattice in ℝn called an algebraic lattice. The center density of Λ is given by
where and denotes the index of the submodule M.
Example 3.1.If, where ζ3 is the primitive 3-th root of unity, thenis a basis of 𝕂 and. If ℳ is a submodule of 𝒪 𝕂 given by
then and
where . Since with and , it follows that
i.e., the center density of σ(ℳ) is the same of the lattice ∧4. Similarly, if, thenis a basis of 𝕂 and. If ℳ is a submodule of 𝒪 𝕂 given by, thenand, where. Sincewithand, it follows that.
Example 3.2.If, whereand ζ9is the primitive 9-th root of unity, thenis an integral basis of 𝕂 and. If ℳ is a submodule of 𝒪 𝕂 given by
then and
where . Since with and , it follows that
i.e., the center density of σ(ℳ) is the same of the lattice Λ3 . Similarly, if, thenand, where. Sincewithand, it follows that. Similarly, if, thenand, where. Sincewithand, it follows that. Finally, if, where θ is a root of, thenis a basis of 𝕂 and. If ℳ is a submodule of 𝒪 𝕂 given by, thenand, where. Sincewithand, it follows that.
Example 3.3.If, where ζ8is the primitive 8-th root of unity, thenis an integral basis of 𝕂 and. If ℳ is a submodule of 𝒪 𝕂 given by
then and
where. Sincewithand, it follows that
i.e., the center density of σ(ℳ) is the same of the lattice Λ4. Similarly, if, thenand, where. Sincewithand, it follows that. Similarly, if, where θ is a root of, then, whereis an integral basis of 𝕂, . If ℳ is a submodule of 𝒪 𝕂 given by, then, and if, then. Sincewithand, if follows that.
Example 3.4.If, whereand ζ44is the primitive 44-th root of unity, thenis an integral basis of 𝕂 and the discriminant of 𝕂 is 114 . Let ℳ be a submodule of 𝒪 𝕂 given by
In this case, ℳ is a submodule of 𝒪 𝕂 of index and the trace form of is given by
Thus,withand. As the volume of the lattice σ (mathcalM) is, it follows that
i.e., the center density of σ(ℳ) is the same of the lattice Λ7 .
Example 3.5.If, where ζ9is the primitive 9-th root of unity, thenis an integral basis of 𝕂 and. If ℳ is a submodule of 𝒪 𝕂 given by
then and
where . Since with and , it follows that
i.e., the center density of σ(ℳ) is the same of the lattice Λ6 .
Example 3.6.If, where ζ20is the primitive 20-th root of unity thenis an integral basis fo 𝕂 and. If ℳ is a submodule of 𝒪 𝕂 given by
then and
where . Since with and , it follows that
i.e., the center density of σ(ℳ) is the same of the lattice Λ8 .
Example 3.7.If, where ζ21is the primitive 21-th root of unity, thenis an integral basis of 𝕂 and. If ℳ is a submodule of 𝒪 𝕂 given by
then and
where . Since with and , it follows that
i.e., the center density of σ(ℳ) is the same of the lattice K12.
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
-
1A.A. Andrade & R. Palazzo Jr. Linear codes over finite rings. TEMA - Trends in Applied and Computational Mathematics, 6(2) (2005), 207-217.
-
2A.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.
-
3A. A. Andrade, A. J. Ferrari & C. W. O. Benedito, Constructions of algebraic lattices, Comput. Appl. Math., 29 (2010) 1-13.
-
4E. Bayer-Fluckiger, Lattices and number fields, In: Contemp. Math., Amer. Math. Soc., Providence (1999), 69-84.
-
5J. H. Conway & N. J. A. Sloane. Sphere Packings, Lattices and Groups, 3rd Edition, Springer Verlag, New York (1999).
-
6J. C. Interlando, J. O. D. Lopes & T. P .N. Neto. The discriminant of abelian number fields, J. Algebra Appl., 5 (2006), 35-41.
-
7P. Samuel. Algebraic Theory of Numbers, Hermann, Paris (1970).
Publication Dates
-
Publication in this collection
30 Apr 2020 -
Date of issue
Jan-Apr 2020
History
-
Received
26 Sept 2019 -
Accepted
01 Jan 2020