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Constructions of Dense Lattices of Full Diversity

ABSTRACT

A lattice construction using ℤ-submodules of rings of integers of number fields is presented. The construction yields rotated versions of the laminated lattices Λn for n = 2, 3, 4, 5, 6, which are the densest lattices in their respective dimensions. The sphere packing density of a lattice is a function of its packing radius, which in turn can be directly calculated from the minimum squared Euclidean norm of the lattice. Norms in a lattice that is realized by a totally real number field can be calculated by the trace form of the field restricted to its ring of integers. Thus, in the present work, we also present the trace form of the maximal real subfield of a cyclotomic field. Our focus is on totally real number fields since their associated lattices have full diversity. Along with high packing density, the full diversity feature is desirable in lattices that are used for signal transmission over both Gaussian and Rayleigh fading channels.

Keywords:
sphere packings; algebraic lattices; number fields; cyclotomic fields

RESUMO

Uma construção de reticulados usando ℤ submódulos de anéis de inteiros de corpos de números é apresentada. A construção produz versões rotacionadas dos reticulados laminados Λn para n = 2, 3, 4, 5, 6, que são os reticulados mais densos nessas dimensões. A densidade de empacotamento esférico de um reticulado é uma função do seu raio de empacotamento, o qual por sua vez pode ser diretamente calculado a partir da norma quadrada mínima do reticulado. Normas em um reticulado realizado por um corpo de números totalmente real podem ser calculadas pela forma traço do corpo restrita ao seu anel de inteiros. Portanto, no presente trabalho, apresentamos também a forma traço do subcorpo real maximal de um corpo ciclotômico. Nosso foco é em corpos de números totalmente reais pois os reticulados associados a eles possuem diversidade máxima. Juntamente com a densidade de empacotamento, a característica de diversidade máxima é desejável em reticulados que são usados para transmissão de sinais que percorrem os canais gaussiano e de desvanecimento Rayleigh.

Palavras-chave:
empacotamento de esferas; reticulados algébricos; corpos de números; corpos ciclotômicos

1 INTRODUCTION

Lattices are discrete subgroups of Euclidean n-space, ℝn , and they have been considered in different applied areas, especially in coding/modulation theory and more recently in cryptography. Algebraic lattices are those obtained via number fields and they have been studied in several papers and from different points of view, see 11 A. A. Andrade & R. Palazzo Jr. Linear codes over finite rings, TEMA - Trends in Applied and Computational Mathematics, 6(2) (2005), 207-217.),(22 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.),(33 R. R. de Araujo, A. C. M. M. Chagas, A. A. Andrade & T. P. Nóbrega Neto . Trace form associated to cyclic number fields of ramified odd prime degree, accepted by J. Algebra App., June, 2019.),(55 E. Bayer-Fluckige. Lattices and number fields, In: Contemp. Math., Amer. Math. Soc., Providence (1999), 69-84.),(66 E. Bayer-Fluckiger . F. Oggier& E. Viterbo. New algebraic constructions of rotated Zn-lattice constellations for the Rayleigh fading channel. IEEE Trans. Inform. Theory, 50(4) (2004) 702-714.),(1010 J. H. Conway & N. J. A. Sloane. “Sphere Packings, Lattices and Groups”, 3rd Edition, Springer Verlag, New York (1999).. These algebraic lattices are constructed through the canonical homomorphism via ℤ -modules of the ring of algebraic integers of a number field. Having the construction of algebraic lattices as our goal, in this paper, we focus on the construction of algebraic lattices with special features, known in the literature, via maximal real subfieds of cyclotomic fields. In digital communications, the lattice parameters of interest are its sphere packing density and minimum product distance. The performance in terms of minimum product distance is given by number field discriminant, i.e., minimizing the discriminant. The question of find totally real number fields with minimal discriminant is a hard problem. Those parameters can be obtained in certain cases of lattices associated to number fields through algebraic properties 66 E. Bayer-Fluckiger . F. Oggier& E. Viterbo. New algebraic constructions of rotated Zn-lattice constellations for the Rayleigh fading channel. IEEE Trans. Inform. Theory, 50(4) (2004) 702-714.. From [44 V. Baustista-Ancora & J. Uc-Kuk. The discriminant of abelian number fields, Journal of Mathematics, 47(1) (2017), 39-52.Thorem 4.1], we can to give a lower (and upper) bound on the minimum product distance. The higher those two parameters are the more attractive the lattice becomes to be used for data transmission over Gaussian and fading channels.

We can find, in literature, several constructions of full diversity rotated ℤn -lattices 55 E. Bayer-Fluckige. Lattices and number fields, In: Contemp. Math., Amer. Math. Soc., Providence (1999), 69-84.) (66 E. Bayer-Fluckiger . F. Oggier& E. Viterbo. New algebraic constructions of rotated Zn-lattice constellations for the Rayleigh fading channel. IEEE Trans. Inform. Theory, 50(4) (2004) 702-714.) (77 E. Bayer-Fluckiger & G. Nebe. On the Euclidian minimum of some real number fields. Journal de Théorie des Nombres de Bordeaux, 17(2) (2005), 437-454.) (88 E. Bayer-Fluckiger & I. S. Atias. Ideal lattices over totally real number fields and Euclidian minima, Archiv der Mathematik, 86(3) (2006), 217-225.. In 66 E. Bayer-Fluckiger . F. Oggier& E. Viterbo. New algebraic constructions of rotated Zn-lattice constellations for the Rayleigh fading channel. IEEE Trans. Inform. Theory, 50(4) (2004) 702-714. it was shown that algebraic lattices obtained via totally real number fields have full diversity. In 88 E. Bayer-Fluckiger & I. S. Atias. Ideal lattices over totally real number fields and Euclidian minima, Archiv der Mathematik, 86(3) (2006), 217-225.the authors presented constructions of algebraic lattices over totally real fields and in 77 E. Bayer-Fluckiger & G. Nebe. On the Euclidian minimum of some real number fields. Journal de Théorie des Nombres de Bordeaux, 17(2) (2005), 437-454.the authors presented constructions of lattices and the trace form for ζr + ζpr-1. In this work, we focus our construction on totally real number fields and we present the trace form for ζn + ζn-1, where n 5, where we present constructions of algebraic lattices of full diversity and optimal packing density. In full diversity algebraic lattices, the minimum product distance depends on the minimum of the algebraic norm of nonzero elements in the lattice. This gives an advantage provided by the algebraic tools, once to calcule the minimum product distance in a general lattice is a hard problem.

In this paper, we present constructions of algebraic lattices of optimal center density and full diversity, that is, we present constructions of algebraic lattices on totally real number fields. In the first construction, the fields are the maximal real subfields of cyclotomic fields, and in the second construction, the fields 𝕂 are extensions of ℚ of degree p where p is an odd ramified prime in 𝒪𝕂, the ring of integers of 𝕂. In Section 2, we review necessary concepts from number fields and lattices. In Section 3, we present explicit trace forms over the maximal totally real subfield of cyclotomic fields. In Section 4, we present constructions of laminated algebraic lattices in dimensions 2 up to 6 with optimal center density. In Section 5 we give our conclusions.

2 BASIC RESULTS OVER ALGEBRAIC LATTICES AND NUMBER FIELDS

In this section, we briefly review the definitions and results that will be needed subsequently. Many of them will be assumed known to the reader; those interested in further details are referred to 1010 J. H. Conway & N. J. A. Sloane. “Sphere Packings, Lattices and Groups”, 3rd Edition, Springer Verlag, New York (1999).), (1414 L. Washington. “Introduction to cyclotomic fields”, Springer-Verlag, New York (1995)..

Let Λ be a full lattice in ℝn , that is, Λ is the set of all integral linear combinations of some basis of the vector space ℝn . Let t denote half the minimal distance between (distinct) lattice points. We can then immediately construct a sphere packing from Λ by centering an n-dimensional sphere with radius t at each lattice point. The obtained arrangement is called the sphere packing associated to Λ. The proportion of the space that is occupied by the spheres is called the sphere packing density of Λ, denoted by ∆(Λ). For comparison purposes, a more used parameter is the center density of the packing, denoted by δ (Λ), which in turn equals ∆(Λ) divided by V n , the volume of an n-dimensional sphere of radius 1.

The minimum product distance of Λ is defined as

d p , m i n Λ = m i n i = 1 n x i : 0 x 1 , . . . , x n Λ .

Lattice Λ is said to be of full diversity if dp, minΛ 0. In the present work, we will only be concerned with constructing lattices with a high sphere packing density and full diversity.

A number field 𝕂 is said to be Abelian (cyclic) if the extension 𝕂/ℚ is Galois and

its Galois group, Gal(𝕂/ℚ), is Abelian (cyclic). Let 𝕂 be a number field of degree n . and signature [r 1 , r 2]. The ℚ-monomorphisms of 𝕂 into ℂ whose images are contained in ℝ are denoted by σ 1 , · · · , σ r1 , and those whose images are not contained in ℝ are denoted by σr1 + 1, σr1 + 1¯, ..., σr1 + r2, σr1 + r2¯ The set OK = α K : there is a monic polynominal fx x such that fα = 0 is called the ring of algebraic integers of 𝕂. If {α 1 , α 2 ,. . . α n } is a ℤ-basis of 𝒪 𝕂, the integer dK = detσjαii, j = 1n2 is an invariant over change of basis and is called the discriminant of 𝕂. The trace of any element x O is defined by TrK/x = i = 1nσix.

If 𝔎 (x) and ℑ(x) denote, respectively, the real and imaginary parts of x K, then the canonical homomorphism σ : K n is defined by

σ x = σ 1 x , . . . , σ r 1 x , R σ r 1 + 1 x , I σ r 1 + 1 x , . . . , R σ r 1 + r 2 x , I σ r 1 + r 2 x ,

for every x K. If is a ℤ-module of 𝕂 of rank n, the set Λ = σM is an n-dimensional lattice in ℝn . If either r1 = 0 or r2 = 0, then the center density of Λ is given by

δ Λ = t n / 2 2 n d K O K : M = t n 2 n d K O K : M ,

where OK : M denotes the index of in 𝒪𝕂, and

t = c k m i n T r K / x x ¯ : x M , x 0

with ck = 1 or 2-1 according to whether r2 = 0 or r1 = 0, respectively. The quantity 2nOK : MdK is equal to the volume of σ ( ). Lattices constructed from totally real number fields, i.e., those with r2 = 0, always have full diversity, a desirable property for practical applications as already observed.

3 TRACE FORMS OVER SUBFIELDS OF CYCLOTOMIC FIELDS

In this section, we present an explicit trace form over maximal real subfield of Q(ζ n ) that allow us to find the packing radius of algebraic lattices. In Subsection 3.1, we present explicit trace forms over the maximal totally real subfield of cyclotomic field Q(ζ n ). Thus, in Theorem 2, we present the trace form for any n, in the Corollaries 1, 2 and 3, for n = pr, n = 2pr and n = pq, respectively, where p and q are prime numbers and r an integer such that r 1. We present it again in the Subsection 3.2, since it will be used in the next section for the constructions of laminated lattices.

3.1 Trace form over maximal real subfield of ℚ(ζ n )

A number field 𝕃 is said to be cyclotomic if L = ζn, where ζ n is a primitive n-th root of unity. Furthermore, L : = φn, where φ is Euler’s phi function, the ring of algebraic integers of 𝕃 is given by QL = ζn and 1, ζn, ζn2, ..., ζnφn - 1 is an integral basis for 𝕃. If K = ζn + ζn-1 is the maximal real subfield of the cyclotomic field ζn, then K : = φn/2, OK = ζn + ζn-1 and ζn + ζn-1, ζn2 + ζn-2, ..., ζnφn2 + ζn-φn2 is an integral basis of 𝒪 𝕂1414 L. Washington. “Introduction to cyclotomic fields”, Springer-Verlag, New York (1995)..

Let n = p1p2...pk + 1, where the p i are distinct prime numbers, a = p1p2...pk and b = pk + 1. Observe that ζa = ζnb and ζbζna, whence ζa, ζb ζn. On the other hand, since gcda, b = 1, one has au + bv = 1 for some u, v . Thus,

ζ n = ζ n a u + b v = ζ n a u ζ n b v = ζ b u ζ a v

and ζn ζa, ζb, that is, ζn = ζa, ζb. Furthermore, ζa ζb = because gcda, b = 1. Since ζn/ is an Abelian extension, it follows that ζn/F and F/, for any F ζn, are Abelian extensions. Finally, the mapping

φ : G a l ζ n / ζ a G a l ζ b / σ σ ζ b

is an isomorphism. Furthermore, if α ζb, then

T r ζ n / ζ a α = T r ζ b / d / α .

Lemma 1. 11 11 J. C. Interlando, T. P. Nóbrega Neto , T. M. Rodrigues & J. O. D. Lopes. A note on the integral trace form in cyclotomic fields, J. Algebra App., 14 (2015), 1550045. Let j and n be integers. If g c d j , n = d , then

T r ζ n / ζ n j = φ n φ n / d T r ζ n / d / ζ n / d j / d .

Lemma 2. 11 11 J. C. Interlando, T. P. Nóbrega Neto , T. M. Rodrigues & J. O. D. Lopes. A note on the integral trace form in cyclotomic fields, J. Algebra App., 14 (2015), 1550045. If i, j and p are integers with i 1 , p prime and g c d j , p = 1 , then

T r ζ n / ζ p i j = - 1 i f i = 1 , 0 i f i > 1 .

Lemma 3.1111 J. C. Interlando, T. P. Nóbrega Neto , T. M. Rodrigues & J. O. D. Lopes. A note on the integral trace form in cyclotomic fields, J. Algebra App., 14 (2015), 1550045. Let n = p1a1p2a2...psas with ak 1 for k = 1, 2, ..., s. If j is prime and gcdj, n = d, then

T r ζ n / ζ n j = φ n φ n / d μ n / d ,

where µ is the Möbius function.

Lemma 4.1111 J. C. Interlando, T. P. Nóbrega Neto , T. M. Rodrigues & J. O. D. Lopes. A note on the integral trace form in cyclotomic fields, J. Algebra App., 14 (2015), 1550045.Letn = p1a1p2a2...psas with ak 1 for k = 1, 2, ..., s. If i is an integer such thati < φn and d = gcdi, n, then

T r ζ n / ζ n i 0 d = n / P t j a n d i = n / P j ,

where P = p1 . . . ps, tj = gcd j, P and j = 1, 2, . . . , φP 1

Theorem 5.1111 J. C. Interlando, T. P. Nóbrega Neto , T. M. Rodrigues & J. O. D. Lopes. A note on the integral trace form in cyclotomic fields, J. Algebra App., 14 (2015), 1550045.Letn= p1a1...psas,with ak 1,for k= 1, 2, . . . ,s, m=φn.If x=a0+a1ζn+ ...+am1ζnm1is an element of ℤ[ζ n ], then

T r ζ n / x x ¯ = n P φ P i = 0 m 1 a i 2 + 2 j = 1 φ P 1 μ P t j φ t j A n P j ,

whereP=p1. . .ps,tj= gcdj, Pfor j= 1,2, . . . ,φP 1, and Ai=a0ai+a1ai+1+. . . +am1iam1for i= 1,2, . . . , m 1. In the next result, we present an explicit trace form over the ring of algebraic integers of (ζn+ζn1).

Theorem 6.LetK=(ζn+ ζn1), where n = p1a1...psas, with aj1for j =1, 2, . . . , sandm=φn. If x=a1(ζn+ζn1)+a2ζn2+ζn2+. . . +aφn2ζnφn2+ζnφn2)is an element ofζn + ζn1, then

T r K / x 2 = m i = 1 m / 2 a i 2 + n P i = u 2 | n P i φ P ρ t i a n 2 P i 2 + 2 i = 1 s ρ t i A n P i + 2 i = v φ P 1 ρ t i B n P i .

whereP = p1. . . ps, ti = gcdi, P,[y]is the greatest integer less than or equal to y, ⌈y⌉ is the smallest integer greater than or equal toy,u=2Pn,s=2Pn1,v=3Pn,ρti= µPtiφti,Aj=a1aj+1+a2aj+2+···+am2jam2 and Bj=k>1k<jkm2akajk. Proof.If xOK, then

x = a 1 ζ n + ζ n 1 + a 2 ζ n 2 + ζ n 2 + ... + a m 2 ζ n m 2 + ζ n m 2

where ai, fori= 1,2, . . . ,φn2. Therefore,

x 2 = a 1 ζ n + a 1 ζ n 1 + ... + a m 2 ζ n m 2 a 1 ζ n + a 1 ζ n 1 + ... + a m 2 ζ n m 2 = a 1 ζ n + a 2 ζ n 2 + ... + a m 2 ζ n m 2 + a 1 ζ n 1 + a 2 ζ n 2 + ... + a m 2 ζ n m 2 2 = A 2 + A ¯ 2 + 2 A A ¯ ,

where A=a1ζn+a2ζn2+...+am2ζnm2. So,

x 2 = j = 1 m / 2 a j 2 ζ n 2 j + ζ n 2 j + 2 j = 3 m 1 B j β j + 2 j = 1 m / 2 a j 2 + j = 1 m / 2 1 A j β j ,

where Aj=a1aj+1+a2aj+2+ · · · +am2jam2,Bj=k>1k>jkm2akajk,andβJ =ζnj+ζnj. Since

T r ζ n / x 2 = ζ n : K T r K / x 2 ,

it follows that

t = T r K / x 2 = 1 2 T r ζ n / x 2 .

Thus,

t = 1 2 T r ζ n / 2 j = 1 m / 2 a j 2 + 2 j = 1 m / 2 1 A j β j j = 1 m / 2 a j 2 ζ n 2 j + ζ n 2 j + 2 j = 3 m 1 B j β j = m j = 1 m / 2 a j 2 + j = 1 m / 2 a j 2 + T r ζ n / ζ n 2 j + 2 j = 1 m / 2 1 A j T r ζ n / ζ n j + j = 3 m 1 B j T r ζ n / ζ n j .

From Lemmas 3 and 4, it follows that

j = 1 m / 2 a j 2 T r ζ n / ζ n 2 j = n P i = u 2 | n P i φ P μ P t i φ t i a n 2 P i 2 , j = 1 m / 2 1 A j T r ζ n / ζ n j = n P i = 1 s μ P t i φ t i A n 2 P i , j = 3 m 1 B j T r ζ n / ζ n j = n P i = v φ P 1 μ P t i φ t i B n 2 P i ,

where u, s, and v are as in the theorem statement. Therefore,

T r K / x 2 = m i = 1 m / 2 a i 2 + n P i = u 2 | n P i φ P ρ t i a n 2 P i 2 + 2 i = 1 s ρ t i A n 2 P i + 2 i = v φ P 1 ρ t i B n 2 P i ,

as desired.

Corollary 7.Letn = pr, with p an odd prime number and r a positive integer. Ifx = a1ζn+ζn1+a2ζn2ζn2+...+aφpr2ζ2φpr2+ζ2φpr2is an element ofζn+ζn1, then

T R K / x 2 = p r i = 1 φ p r 2 a i 2 p r 1 i = u 2 | π p i p 1 a i p r 1 2 2 + 2 i = 1 p 3 2 A i p r 1 + 2 i = v p 2 B i p r 1 ,

whereu=2pr1,v=3pr1,Aj=a1aj+1+a2aj+2+...+aφpr2jaφpr2andBj=k1k<jkφpr2akajkProof. Since P=p, it follows that nP=pr1andφP =p 1. Thus, from Theorem 6, it follows that u=2pr1,s=p32, andv=2pr1.Now,ti= gcdi, P = gcdi,p = 1, because 1 ip 1, hence ρti =µPtiφti =µpφ1 = 1. The result now follows from Theorem 6. ◻

Corollary 8.Letn=2pr, where p is an odd prime number and r is a positive integer. Ifx=a1ζn+ζn1+a2ζn2+ζn2+...+aφ2pr2ζnφ2pr2+ζnφ2pr2is an element ofζn+ζn1, then

T r K / x 2 = φ 2 p r i = 1 φ 2 p r 2 a i 2 p r 1 i = u 2 | i p 1 a i p r 1 2 2 2 U + 2 V ,

whereU=i=12|ip32Aipr1+i=v2|ip2Bipr1,V=i=12|ip32Aipr1+i=v2|ip2Bipr1,u=2pr1,v=3pr1,Aj=a1aj+1a2aj+2+...+aφ2pr2jaφ2pr2,Bj=k1k<jkφ2pr3akajkProof. Since P= 2p, it follows that np=pr1,andφP=p1. From Theorem 6, it follows that u=2pr1,s=p32andv=3pr1. Also,

t i = gcd i , P = gcd i ,2 p = 1 if i is odd , 2 if i is even .

Therefore,

ρ t i = 1 if i is odd , -1 if i is even .

Since p is odd, it follows that nP=pr1 odd. So, nPii is even if and only if i is even. The result now follows from Theorem 6. ◻

Corollary 9. Let n = p q , where p and q are distinct primes. If x = a 1 ζ n + ζ n 1 + a 2 ζ n 2 + ζ n 2 + ... + a φ p q 2 ζ n φ p q 2 + ζ n φ p q 2 is an element of ζ n + ζ n 1 , then

T r K / x 2 = φ p q i = 1 φ p q 2 a i 2 + U + 2 V + 2 W ,

where

s = φ p q 2 - 1 , A j = a 1 a j + 1 + a 2 a j + 2 + . . . + a φ p q 2 - j a φ p q 2 , B j = k 1 k < j - k φ p q 2 a k a j - k , U = - p - 1 i = 2 2 p | i φ p q a i 2 2 - q - 1 i = 2 2 q | i φ p q a i 2 2 + i = 2 2 | i , g c d i , p q = 1 φ p q a i 2 2 , V = - p - 1 i = 1 p | i s A i - q - 1 i = 1 q | i s A i + i = 1 g c d i , p q = 1 s A i , W = - p - 1 i = 3 p | i φ p q B i - q - 1 i = 3 q | i φ p q - 1 B i + i = 3 g c d i , p q = 1 φ p q - 1 B i ,

Proof. From Theorem 5, m=φpq=φP,µP=µpq = 1, andtj= gcdj, P with j= 1,2, . . . ,φpq 1. Thus,

t j =gcd(j ,pq)= p if j is a multiple of p q if j is a multiple of q 1 otherwise ,

and

p t i = - p-1 if j is a multiple of p - q-1 if j is a multiple of q 1 otherwise .

Furthermore, nPj=j. Thus, if n=pq and x=a1(ζn+ζn1)+a2ζn2+ζn2+...+am2ζnm2+ζnm2 is an element of ℤ [ζn ]. The result now follows from Theorem 6.

3.2. Trace form over cyclic number fields of degree p

In 1313 E. L. Oliveira, J. C. Interlando , T. P. da Nóbrega Neto & J. O. D. Lopes. The integral trace form of cyclic extensions of odd prime degree, Rocky Mountain J. Math., 47 (2017), 1075-1088., a lattice construction using cyclic number fields of degree p, where p is unramified, was presented. Lattices whose center densities are arbitrarily close to optimal values were obtained. Below we present a similar construction, in this case using cyclic number fields of degree p, where p is ramified. Let 𝕂 be a cyclic number field of prime degree p >2. From KroneckerWeber Theorem 1414 L. Washington. “Introduction to cyclotomic fields”, Springer-Verlag, New York (1995)., it follows that there exists n >0 such that K(ζn). The least integer n with the property K(ζn) is called the conductor of 𝕂. In this case, the discriminant of 𝕂 is given by dK=np1 [9, p. 186]. If n is the conductor of a cyclic number field 𝕂 of odd prime degree p, then

  1. p is ramified in 𝕂 if and only if n=p2orn=p2p1p2. . .pr ;

  2. p is unramified in 𝕂 if and only if n=p1p2. . . pr ,

where r1 and the pi are distinct prime numbers such that pi1(modp).

Let 𝕂 be a cyclic number field of degree p and conductor n=p2orn=p2p1p2. . .pr, where the p i are distinct prime numbers such that pi1(modp). From 1212 G. Lettl. The ring of integers of an Abelian number field, J. Reine Angew. Math., 404 (1990), 162-170. , if t=Tr(ζn)/K(ζn), then

  1. K=t .

  2. B= {1,σt,· · ·,σp1t} is a ℤ-basis for 𝒪𝕂.

Theorem 10.33 R. R. de Araujo, A. C. M. M. Chagas, A. A. Andrade & T. P. Nóbrega Neto . Trace form associated to cyclic number fields of ramified odd prime degree, accepted by J. Algebra App., June, 2019.)Let 𝕂 be a cyclic number field of prime degreep >2,K(ζn), where n=p2p1p2. . .prwith r 1and the p i distinct prime numbers such thatpi 1 modp. If x=a0+Σi=1p1aiσitOK, where ai, for i= 1,2,. . ., p 1, then

T r K / Q x 2 = p a 0 2 + p p 1 ... p r 2 1 i j p 1 a i a j + p 1 i = 1 p 1 a i 2 = p a 0 2 + p p 1 ... p r i = 1 p 1 a i 2 + 1 i j p 1 a i a j 2 .

4 CONSTRUCTIONS OF LAMINATED ALGEBRAIC LATTICES OVER NUMBER FIELDS

In this section, we explicit constructions of algebraic densest lattices in dimensions 2 up to 6. The strategy used is to search for submodules contained in the ring of algebraic integers that perform the laminated lattices via canonical homomorphism. In this sense, we use center density as a parameter, so the trace forms (Theorem 6 and Theorem 10) are important for calculating the packing radius.

4.1 The Λ2-laminated lattice

If K=(ζ12+ζ121), then [K:] = 2, {ζ12+ζ121,ζ122+ζ122} is an integral basis for 𝕂 and dK= 12. If

M = a 1 ζ 12 + ζ 12 1 + a 2 ζ 12 2 + ζ 12 2 O K : a 1 + a 2 0 m o d 2 ,

then is a ℤ-submodule of 𝒪 𝕂 of rank 2 and index 2. From Theorem 6, the trace form of αM is given by

T r K / ( α 2 ) = 8 a 1 2 a 1 a 2 + a 2 2 .

Thus, t= min{TrK/(α2) :αM,α 0}=8, which is attained at a1= 1 and a2= 0. Since the volume of lattice σ(M) equals 22dK[M:OK] = 243, it follows that

( δ ( σ ( M ) ) = 2 3 2 2 4 3 = 1 2 3 ,

i.e., the center density of σ (ℳ) is the same as that of lattice Λ2.

4.2 The Λ3-laminated lattice

If K=ζ9+ζ91, then K: = 3, ζ9+ζ91,ζ92+ζ92,ζ93+ζ93 is an integral basis for K and dK= 34. If

M = a 1 ζ 9 + ζ 9 1 + a 2 ζ 9 2 + ζ 9 2 + a 3 ζ 9 3 + ζ 9 3 O K : 4 a 1 + 4 a 2 + a 3 0 m o d 6 a n d a 3 0 m o d 2 ,

then ℳ is a ℤ-submodule of 𝒪 𝕂 of rank 3 and index 6. From Theorem 6, the trace form of αM is given by

T r K / α 2 = 18 a 1 2 + a 1 a 2 + 4 a 1 a 3 + a 2 2 4 a 2 a 3 + 2a 3 2

Thus, t= min{TrK/(α2) :αM,α 0} = 18, which is attained at a1= 1 and a2=a3= 0. Since the volume of lattice σ(M) equals 23dK[M:OK] = 2433, it follows that

δ σ M = 2 3 2 3 2 4 3 3 = 1 4 2 ,

i.e., the center density of σ ( ) is the same as that of lattice Λ3.

4.3 The Λ4-laminated lattice

If K=ζ24+ζ241, then K: = 4, ζ24+ζ241,ζ242+ζ242,ζ243+ζ243,ζ244+ζ244 is an integral basis for K and dK= 28. 32. If

M = a 1 ζ 24 + ζ 24 1 + a 2 ζ 24 2 + ζ 24 2 + a 3 ζ 24 3 + ζ 24 3 + a 4 ζ 24 4 + ζ 24 4 O K : 4 a 1 + 3 a 2 + 2 a 3 + a 4 0 m o d 6 ,

then is a submodule of 𝒪 𝕂 of rank 4 and index 6. From Theorem 6, the trace form of αM is given by

T r K / α 2 = 24 a 1 2 + 2 a 1 a 2 + 3 a 1 a 3 + 4 a 1 a 4 + 2 a 2 2 + 4 a 2 a 3 + 6 a 2 a 4 + 3 a 3 2 + 8 a 3 a 4 + 6 a 4 2 .

Thus, t= min{TrK/(α2) :αM,α 0} = 24, which is attained at a0= 1 and a1=a2=a3= 0. Since the volume of lattice σ(M) equals 24dK[M:OK] = 29. 32, it follows that

δ σ M = 2 3 3 4 2 9 3 2 = 1 8 ,

i.e., the center density of σ ( ) is the same as that of the lattice Λ4.

4.4 The Λ5-laminated lattice

Let K be a number field of degree p= 5 and conductor n= 52. In this case, the Galois group GalK/ =σ is cyclic of order 5, t=Trζ52/Kζ52, and dK= 58. Let be the submodule of 𝒪 𝕂 of rank 5 and index 10 given by

M = a 0 + a 1 σ t + a 2 σ 2 t + a 3 σ 3 t + a 4 σ 4 t O K : a 0 0 m o d 2 , a 0 + a 1 + a 2 + a 3 + a 4 0 m o d 5 .

From Theorem 10, it follows the trace form of K restricted to is given by

T r K / x 2 = 50 2 x 0 2 + 6 x 0 x 1 + 6 x 0 x 2 + 6 x 0 x 3 + 8 x 0 x 4 + 6 x 1 2 + 11 x 1 x 2 + 11 x 1 x 3 + 15 x 1 x 4 + 6 x 2 2 + 11 x 2 x 3 + 15 x 2 x 4 + 6 x 3 2 + 15 x 3 x 4 + 10 x 4 2 ,

where x 0 , . . . , x 4 are any integers. It follows that t= min {TrK/x2 :xM} = 50 is attained at a0=a1=a2= 0 and a3= a4= 1. Since the volume of lattice σ(M) equals 25dKM:OK= 26 55, one has

δ σ M = 2 5 2 5 2 6 5 5 = 1 8 2 ,

i.e., the center density of σ () equals that of lattice Λ5.

4.5 The Λ6-laminated lattice

If K=ζ36+ζ361, then K: = 6, ζ36+ζ361,ζ362+ζ362,ζ363+ζ363,ζ364+ζ364,ζ365+ζ365,ζ366+ζ366 is a basis of K and dK= 2639. IfM=2+ζ36+ζ36 ζ36+ ζ364 ζ365+ζ365OK. In this case, is a submodule of 𝒪 𝕂 of rank 6 and index 72. From Theorem 6, the trace form of αM is given by

T r K / α 2 = 72 ( a 0 2 + 2 a 0 a 1 2 a 0 a 3 2 a 0 a 4 a 0 a 5 + 2 a 1 2 2 a 1 a 3 3 a 1 a 4 + a 2 2 + a 2 a 3 + 2 a 2 a 4 + a 2 a 5 + 3 a 3 2 + 5 a 3 a 4 + 2 a 3 a 5 + 3 a 4 2 + 2 a 4 a 5 + a 5 2 .

Thus, t= min{TrK/(α2) :αM,α 0} = 72 which is attained at a0= 1 and a1=a2=a3=a4=a5= 0. Since volume of the lattice σ(M) equals 26dKM:OK = 212· 363 it follows that

δ M = 2 3 3 2 6 2 12 3 6 3 = 1 8 3 ,

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

5 CONCLUSION

A construction of algebraic lattices with special features, namely, high center density and full diversity, was presented. Each lattice was obtained as the image of the canonical homomorphism from a suitably chosen ℤ -submodule of the ring of integers of the maximal real subfield of a cyclotomic field into ℝn (n-dimensional Euclidean space). The trace form of the maximal real subfield of the cyclotomic field ℚ(ζ n ) was derived explicitly so that the minimum of the associated lattice could be determined. As a result, rotated versions of full diversity of laminated lattices in dimensions 2 to 6 have been obtained. Although the constructed lattices are well known, this work helps answer the question of which lattices can be realized by a given number field, as posed in 55 E. Bayer-Fluckige. Lattices and number fields, In: Contemp. Math., Amer. Math. Soc., Providence (1999), 69-84.. Whether the presented technique can be used to yield higher dimensional lattices with the desired features (full diversity and high packing density) is left as a research problem. Calculating or providing a good lower bound for the minimum product distance66 E. Bayer-Fluckiger . F. Oggier& E. Viterbo. New algebraic constructions of rotated Zn-lattice constellations for the Rayleigh fading channel. IEEE Trans. Inform. Theory, 50(4) (2004) 702-714. of the lattices obtained by the construction technique of this work is also left as a research problem.

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.

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Publication Dates

  • Publication in this collection
    03 Aug 2020
  • Date of issue
    May-Aug 2020

History

  • Received
    13 June 2019
  • Accepted
    19 Mar 2020
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