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A Construction of Rotated Lattices via Totally Real Subfields of the Cyclotomic Field (ζp)

ABSTRACT

The theory of lattices have shown to be useful in information theory and rotated lattices with high modulation diversity have been extensively studied as an alternative approach for transmission over a Rayleigh-fading channel, where the performance of this modulation schemes essentially depends on the modulation diversity and on the minimum product distance to achieve substantial coding gains. The maximum diversity of a rotated lattice is guaranteed when we use totally real number fields and the minimum product distance is optimized by considering fields with minimum discriminant. In this paper, we present construction of a full diversity rotated lattice for the Rayleigh fading channel in Euclidean space with full diversity, where this construction is through a totally real subfield 𝕂 of the cyclotomic field Q(ζ p), where p is an odd prime, obtained by endowing their ring of integers.

Key words
lattices; cyclotomic fields; algebraic number field; rotated lattice

RESUMO

A teoria de reticulados têm mostrado útil na teoria da informação e reticulados ideais com alta diversidade de modulação têm sido extensamente estudados como uma alternativa de transmissão via o canal de Rayleigh, onde o desempenho destes esquemas de modulação depende essencialmente da diversidade de modulação e da distância produto mínima para obter ganhos substanciais de codificação. A diversidade máxima de um reticulado rotacionado é garantida quando usamos corpos de números totalmente reais e a distância produto mínima é otimizada considerando os corpos com discriminante mínimo. Neste trabalho, apresentamos uma construção de reticulado rotacionado, onde esta construção é através de um subcorpo totalmente real 𝕂 do p-ésimo corpo ciclotômico, onde p é um número primo ímpar, obtido via o seu anel de inteiros.

Palavras-chave
reticulados; corpos ciclotômicos; corpos de números algébricos; reticulado rotacionado.

1 INTRODUCTION

Algebraic number theory has recently raised a great interest for their new role in algebraic lattice theory and for application in coding and modulation. The problem of finding algebraic lattices with maximal minimum product distance has been studied in last years and this has motivated special attention of many researchs in considering ideals of certain rings [5][5] A. de Andrade & R. Palazzo Jr. Linear codes over finite rings. TEMA - Trends in Applied and Computational Mathematics, 6(2) (2005), 207–217., [2][2] A. Ansari, T. Shah, Z.u. Rahman & A.A. Andrade. Sequences of Primitive and Non-primitive BCH Codes. TEMA - Trends in Applied and Computational Mathematics, 19(2) (2018), 369–389. and [1][1] A.A. Andrade & J.C. Interlando. Rotated ℤn-lattices via real subfields of ℚ(ζ2r). TEMA - Trends in Applied and Computational Mathematics, 20(3) (2019).. Eva Bayer et al. [8][8] F. Oggier, E. Bayer-Fluckiger & E. Viterbo. New algebraic constructions of rotated cubic lattice constellations for the Rayleigh fading channel. In “Proceedings 2003 IEEE Information TheoryWorkshop (Cat. No. 03EX674)”. IEEE (2003), pp. 263–266. and Andrade et al. [1][1] A.A. Andrade & J.C. Interlando. Rotated ℤn-lattices via real subfields of ℚ(ζ2r). TEMA - Trends in Applied and Computational Mathematics, 20(3) (2019). have presented families of rotated n-lattices based on algebraic number theory. We know that totally real algebraic number fields result in the maximum diversity, equal to the dimension of the lattice [3][3] J. Boutros, E. Viterbo, C. Rastello & J.C. Belfiore. Good lattice constellations for both Rayleigh fading and Gaussian channels. IEEE Transactions on Information Theory, 42(2) (1996), 502–518.. This motivates the investigation on lattices over totally real number fields.

A lattice Λ is a discrete additive subgroup of n, equivalently, Λn is a lattice iff there are linearly independent vectors v1,v2,,vmn such that

Λ={i=1maivi:ai,fori=1,2,,m}.
The set {v1,v2,,vn} is a -basis and a matrix M whose rows are these vectors is said to be a generator matrix for Λ and the associated Gram matrix is given by G=MMt=(<vi,vj>)i,j=1n. Lattices have been considered in different areas, especially in coding theory and more recently in cryptography. In this paper, we attempt to construct lattices with full rank, i.e., m=n, which may be suitable for signal transmission over both Gaussian and Rayleigh fading channels [3][3] J. Boutros, E. Viterbo, C. Rastello & J.C. Belfiore. Good lattice constellations for both Rayleigh fading and Gaussian channels. IEEE Transactions on Information Theory, 42(2) (1996), 502–518.. For this purpose the tattice parameters we consider here are diversity and minimum product distance.

In [1][1] A.A. Andrade & J.C. Interlando. Rotated ℤn-lattices via real subfields of ℚ(ζ2r). TEMA - Trends in Applied and Computational Mathematics, 20(3) (2019)., for any integer r4, rotated n-lattices, n=2r2 and n=2r3, were constructed from (ζ2r+ζ2r1), the maximal real subfield of (ζ2r), and over (ζ2r2+ζ2r2), where ζ2r is a primitive 2r-th root of unity. In this work, having the construction procedure of a rotated lattice over the maximal real subfield of a cyclotomic field as the main motivation, we make use of algebraic number theory for constructing rotated lattices via totally real subfields of the cyclotomic field (ζp), where p is an odd prime number.

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 totally real subfields of the cyclotomic field (ζp), where p is an odd prime number. In Section 4, an algorithm to the construction of rotated lattices is presented and we present examples in terms of center density and minimum product distance.

2 BASIC RESULTS FROM NUMBER THEORY

A number field is a field 𝕃 that is a finite degree extension n of . An element α𝕃 is called an algebraic integer if there is a monic polynomial f(x) with integer coefficients such that f(α)=0. The set

𝒪𝕃={α𝕃:αis an algebraic integer}
is a ring called ring of algebraic integers of 𝕃. The ring 𝒪𝕃 is a -module of rank n and a -basis {α1,,αn} of 𝒪𝕃 is called an integral basis of 𝕃 (or of 𝒪𝕃). Furthermore, 𝕃=(α), where α is a root of a monic irreducible polynomial p(x)[x]. The n distinct roots α1,α2,,αn of p(x) are the conjugates of α. If σi:𝕃 is a -homomorphism, then σi(α)=αi for some i=1,2,,n, and there are exactly n -homomorphism σi:𝕃, for i=1,2,,n. A homomorphism σi is said to be real if σi(𝕃) and imaginary otherwise. A number field 𝕃 is said to be totally real if σi is real for all i=1,2,,n and totally imaginary if σi is imaginary for all i=1,2,,n. The trace of any element α𝕃 is defined as the rational number
T𝕃:(α)=i=1nσi(α),
and if α𝒪𝕃, then T𝕃:(α). The discriminant of L, denoted by Δ𝕃, is the rational integer given by det(Tr𝕃:(αiαj)).

A cyclotomic field is a number field 𝕃 such that 𝕃=(ζn), where ζn is a primitive n-th root of unity. Also, [𝕃:]=φ(n), where φ is the Euler function, 𝒪𝕃=[ζn] is the ring of algebraic integers of [ζn], and the field 𝕂=(ζn+ζn1) is the maximal real subfield of 𝕃, where [𝕃:𝕂]=2 and 𝒪𝕂=[ζn+ζn1] [6][6] B. Erez. The Galois structure of the trace form in extensions of odd prime degree. Journal of Algebra, 118(2) (1988), 438–446., [9][9] P. Ribenboim. “Classical theory of algebraic numbers”. Springer Science & Business Media (2013)..

3 CONSTRUCTION OF A ROTATED LATTICE

If ζp is a primitive p-th root of unity, where p is an odd prime number, then is a cyclic extension of degree p1 over that contains the real subfield (ζp+ζp1), which is cyclic of degree l=(p1)/2 over . If G=Gal(𝕃:) is the Galois group (cyclic) of 𝕃 over with generator σr (or σ), then σr(ζp)=ζpr, where r is a generator of p*, and rl1(modp), that is, r is a primitive element modulo p.

Theorem 1. [11][11] J.P. Serre. “A course in arithmetic”, volume 7. Springer Science & Business Media (2012). (Dirichlet’s theorem) If a,n are integers such that 1an and gcd(a,n)=1, then the arithmetic progression {a,a+n,a+2n,,a+kn,} contains infinitely many primes.

If n is a positive integer, from Theorem 1, it follows that there exists a prime p such that p1(modn). Since n divides p1, from Galois Correspondence Theorem, it follows that there exists a unique field 𝕂 contained in (ζp) which is cyclic of degree n over . If n is an even number that divides (p1)/2 or if n is an odd number, then 𝕂 is contained in the real subfield (ζp+ζp1). In this case, 𝕂=(θ), where θ=Tr𝕃:𝕂(ζp),

Gal((ζp):𝕂)=σn={σn,σ2n,,σnm}={σrn,σr2n,,σrmn},
where m=(p1)/n, Gal(𝕂:)={σ0,σ,,σn1} and
{σr(θ),σr2(θ),,σrn(θ)}={θ,σ(θ),,σn1(θ)}
is an integral basis of 𝕂, where σs=σrs, for all s+ [6][6] B. Erez. The Galois structure of the trace form in extensions of odd prime degree. Journal of Algebra, 118(2) (1988), 438–446..

If σ𝕂 is the canonical embedding given by

σ𝕂:𝕂nxσ𝕂(x)=(x,σ(x),,σn1(x)),
then σ𝕂(𝒪𝕂) is an algebraic lattice in n with maximum diversity. Since the set {θ,σ(θ),,σn1(θ)} is a -basis of 𝒪𝕂, it follows that
{σ𝕂(θ),σ𝕂(σ(θ)),,σ𝕂(σn1(θ))}n
is a basis of the lattice σ𝕂(𝒪𝕂), whose generator matrix is given by

M=(θσ(θ)σn1(θ)σ(θ)σ2(θ)θσn1(θ)θσn2(θ)).
Since M=Mt, it follows that the i-th row is given by
σ𝕂(σi(θ))=(σi(θ),σi+1(θ),,σi+n1(θ)),
for i=0,1,,n1. The Gram matrix G=(gij)i,j=0n1 of σ𝕂(𝒪𝕂) is given by G=MMt, where
gij=σ𝕂(σi(θ)),σ𝕂(σj(θ))=σi(θ)σj(θ)+σi+1(θ)σj+1(θ)++σi+n1(θ)σj+n1(θ)=a=0n1σi+a(θ)σj+a(θ)
Since σn|𝕂=σ𝕂0 and σns|𝕂=σs|𝕂, for all s+, it follows that
gij=a=0n1σi+a(θ)σj+a(θ)=a=0n1σa(θ)σj+ai(θ)=a=0n1σa(θσji(θ))=Tr𝕂:(θσji(θ)),
for i,j=0,1,,n1. Thus,
G=(Tr𝕂:(θσji(θ)))i,j=0n1.
Since Tr𝕂:(σi(θ)σj(θ))=Tr𝕂:(θσji(θ)), for i,j=0,1,,n1, it is sufficient to calculate Tr𝕂:(θσt(θ)), for t=0,1,,n1. Finally,
Tr𝕂/:mathbbQ(θ)=Tr𝕂:(Tr(ζp):𝕂(ζp))=Tr(ζp):(ζp)=1
and
Tr𝕂:(σt(θ))=a=0n1σa(σt(θ))=a=0n1σt(σa(θ))=σt(a=0n1σa(θ))==σt(Tr𝕂:(θ))=σt(1)=1,
for t=0,1,,n1.

The following theorem, which is the main result of this work, gives us the key to constructing full diversity rotated lattice bssed on real subfields of the cyclotomic field (ζp).

Theorem 2. If θ=Tr(ζp):𝕂(ζp), then

T r 𝕂 : ( θ σ t ( θ ) ) = { p ( p 1 n ) if t = 0 ; ( p 1 n ) if t = 1 , 2 , , n 1 .

Proof. Since θ𝕂, it follows that

Tr𝕂:(θσt(θ))=a=0n1σa(θσt(θ)),
for all t=0,1,,n1., and
θσt(θ)=c=1p1nσcn(ζp)j=1p1nσt+jn(ζp)=c,j=1p1nσcn(ζp)σt+jn(ζp),
because Gal((ζp):𝕂)=σn, whose order is m=(p1)/n. Thus,
Tr𝕂:(θσt(θ))=a=0n1c,j=1p1nσa+cn(ζp)σa+t+jn(ζp)=a=0n1c,j=1p1nζpra+cnζpra+t+jn=a=0n1c,j=1p1nζpra+cn+ra+t+jn.
Since r is a generator of p*, it follows that rp11(modp), and thus, r(p1)q=(rp1)q1q=1(modp), for all q. So,
ra+(mq+c)n=ra+(p1nq+c)n=ra+(p1)q+cn=ra+cnr(p1)qra+cn(modp).
Therefore, ζpra+cn=ζpra+(mq+c)n, for all q+. Now, if sc(modm), then s=mq+c, for some q. Thus, ζpra+sn=ζpra+cn. So,
Tr𝕂:(θσt(θ))=a=0n1c,jmζpra+cn+ra+t+jn=cma=0n1jmζpra+cn+ra+t+jn.
Furthermore, if dcj(modm), i.e., cd+j(modm), then ζpra+cn=ζpra+(d+j)n, and since c ranges in m, it follows that d also ranges in m. Thus,
Tr𝕂:(θσt(θ))=dma=0n1jmζpra+(d+j)n+ra+t+jn=dma=0n1jmζp(rdn+rt)ra+jn=d=1p1na=0n1j=1p1nζp(rdn+rt)ra+jn.
Since a{0,1,,n1} and j{1,,p1n}, it follows that ra+jns(modp), where s=1,,p1, because r=p*={1,,p1}, and thus ra+jn=s for some s=1,,p1. So, ζpra+jn=ζps, for some s=1,,p1. Now, since n(p1n)=p1, it follows that
a=0n1j=1p1nζp(rdn+rt)ra+jn=a=0n1j=1p1n(ζpra+jn)rdn+rt=s=1p1(ζps)rdn+rt=s=1p1(ζprdn+rt)s.
Thus, if ωd,t=ζprdn+rt, then
Tr𝕂:(θσt(θ))=d=1p1ns=1p1(ωd,t)s,
where
s=1p1(ωd,t)s={p1ifωd,t=11ifωd,t1,
for t=0,1,,n1. The first case is trivial. Now, for ωd,t1, is sufficient observe that ωd,t=ζprdn+rt is a root of the polynomial
xp1x1=xp1++x+1,
and therefore,
s=1p1(ωd,t)s=(ωd,t)p1++(ωd,t)2+ωd,t=1.
Now, to calcule Tr𝕂:(θσt(θ)), we consider the cases t=0 and t0. But,
ωd,t=1t=0andd=p12n.(3.1)
In fact, if t=0 and d=(p1)/2n, then rp11(modp). Since p1 is even, it follows that there exists l such that p1=2l. So, (rl)2=r2l1(modp), i.e., p(rl)21=(rl+1)(rl1). Thus, rl1(modp) or rl1(modp). But, since the first case is not possible because p1 is the smallet positive integer with this property, it follows that rl1(modp). Thus, rp12+10(modp), and therefore,
ωd,t=ζprdn+1=ζpr(p12n)n+1=ζprp12+1=1.
Reciprocally, if ωd,t=1, i.e., ζprdn+rt=1, then
rdn+rt0(modp)rdnrt(modp).
Since rl+trt(modp), it follows that
rt(modp)rdnrl+t(modp).
From [10][10] J.P.O. Santos. “Introduction to numbers theory, Projeto Euclides”. Impa (2006)., it follows that
rl+t(modp)l+tdn(modp1).
Thus p1 divides l+tdn, i.e., there exists k1 such that t=dnl+k1(p1). Now, n|dn, n|k1(p1) (because n|p1) and n|l (because n(p12n)=l), and thus, n|t. Since t=0,1,,n1, it follows that t=0. Thus, dn=lk1(p1), and therefore,
d=p12nk1(p1n)=p12n(12k1)=k2(p12n),withk2=12k1odd.
Since k2 is positive, because if k2<0, then d0. Thus, k2=1 or 2, because if k23, then d>(p1)/n. But, since k2 is odd, it follows that k2=1. Therefore, d=(p1)/2n, which concludes the proof of the equivalency of the Equation (3.1). Observe that the number p12n is integer because n(p1)/2. Now, for t0, from equivalency of the Equation (3.1), it follows that ωd,t1, and therefore, s=1p1(ωd,t)s=1. Thus,
Tr𝕂:(θσt(θ))=d=1p1ns=1p1(ωd,t)s=d=1p1n1=(p1n).
Now, suppose t=0. From equivalency of the Equation (3.1), if d=(p1)/2n, then ωd,t=1, and if d(p1)/2n, then ωd,t1. Therefore,
Tr𝕂:(θσt(θ))=d=1p1ns=1p1(ωd,t)s=(p1)+d=1,dp12np1n1=(p1)(p1n1)=p(p1n).
Since σji ranges in σt, with t=0,1,,n1, it follows that
Tr𝕂:(θσt(θ))={p(p1n)ift=0(p1n)ift=1,2,,n1,
which concludes the proof. ""²

4 AN ALGORITHM OF CONSTRUCTION OF A ROTATED LATTICE

In this section, we present an algorithm to construct of a rotated lattice and we analyze if these lattices have good performance in terms of center density and minimum product distance. For this, we consider 𝕂 a field such that 𝕂(ζp+ζp1), where p is a prime, [(ζp):𝕂]=m and [𝕂:]=n.

4.1 Algorithm

An algorithm to construct of a rotated lattice is given by:

  1. Choose a dimension n.

  2. Compute a prime p such that p1(modn), where n is an even number that divides (p1)/2 or if n is an odd number.

  3. Compute r such that r is a primitive element modulo p, i.e., r is a generator of p*.

  4. Compute θ=Tr(ζp):𝕂(ζp) and σi(θ), for i=1,,n1, with σ=Gal((ζp):).

  5. Compute the Gram matrix G=(gij)i,j=1n, where

    gi,j=Tr𝕂:(θσji(θ))={p(p1n)ifi=j(p1n)ifij,
    for i,j=1,,n.

4.2 Center density and minimim distance product

If α𝒪𝕂, where α=a0σ0(θ)+a1σ(θ)++an1σn1(θ), then

Tr𝕂:(α2)=i=0n1ai2+m0i<jn1(aiaj)2.
If is a -submodule in 𝕂 of rank n, then the set Λ=σ𝕂() is a lattice in n called an algebraic lattice. The center density of Λ is given by
δ(Λ)=tn/22n[𝒪𝕂:]Δ𝕂,
where t=min{Tr𝕂:(α2):α,α0}, [𝒪𝕂:] denotes the index of the submodule and Δ𝕂=pn1 [7][7] J.C. Interlando, J.O.D. Lopes & T.P. Nóbrega. The discriminant of abelian number fields. Journal of Algebra and its Applications, 5(01) (2006), 35–41.. If x=(x1,,xn)n is an element of Λ, the product distance of x from the origin is defined as
dp(x)=i=1nxi,
and the minimum product distance of Λ is defined as
dp,min(Λ)=minxΛ,x0dp(x).
If is a principal ideal of 𝒪𝕂, then the minimum product distance of Λ is given by
dp,min(Λ)=det(Λ)Δ𝕂,
where det(Λ)=detG [8][8] F. Oggier, E. Bayer-Fluckiger & E. Viterbo. New algebraic constructions of rotated cubic lattice constellations for the Rayleigh fading channel. In “Proceedings 2003 IEEE Information TheoryWorkshop (Cat. No. 03EX674)”. IEEE (2003), pp. 263–266.. The normalized minimum product distance of Λ, dp,norm(Λ), is the minimum product distance of the rotated lattice 1det(G)2nΛ. Thus, the normalized minimum product distance of Λ is given by
dp,norm(Λ)=1(k)n1det(G)dp,min(Λ)=1(k)n1Λ,
where k=min{x2:0xΛ}. Thus,
dp,norm(Λ)n=1k1Λ2n.

4.3 Example

If 𝕃=(ζ5) and 𝕂=(θ), where θ=ζ5+ζ51, then n=2, t=2, Δ𝕂=5,

G=(3223)
is the Gram matrix of the algebraic lattice σ(𝒪𝕂), k=2 and det(G)=5. In this case, the center density is given by δ(Λ)=1/(25) and the normalized minimum product distance. Since k=2, it follows that dp,norm(Λ)=121540.47287. In the Table 1, we summarized a comparision of the values of δ(), where =𝒪𝕂, and dp,norm(Λ)n for some known constructions of algebraic lattices in some dimensions.

Table 1
Comparison of the values of δ() [4][4] J.H. Conway & N.J.A. Sloane. “Sphere Packings, Lattices and Groups”. Springer-Verlag (1998). and dp,norm(Λ)n [8][8] F. Oggier, E. Bayer-Fluckiger & E. Viterbo. New algebraic constructions of rotated cubic lattice constellations for the Rayleigh fading channel. In “Proceedings 2003 IEEE Information TheoryWorkshop (Cat. No. 03EX674)”. IEEE (2003), pp. 263–266..

5 ACKNOWLEDGMENTS

The authors would like to thank the anonymous reviewers for their intuitive comentary that significantly improved the worth of this work.

REFERENCES

  • [1]
    A.A. Andrade & J.C. Interlando. Rotated n-lattices via real subfields of (ζ2r) TEMA - Trends in Applied and Computational Mathematics, 20(3) (2019).
  • [2]
    A. Ansari, T. Shah, Z.u. Rahman & A.A. Andrade. Sequences of Primitive and Non-primitive BCH Codes. TEMA - Trends in Applied and Computational Mathematics, 19(2) (2018), 369–389.
  • [3]
    J. Boutros, E. Viterbo, C. Rastello & J.C. Belfiore. Good lattice constellations for both Rayleigh fading and Gaussian channels. IEEE Transactions on Information Theory, 42(2) (1996), 502–518.
  • [4]
    J.H. Conway & N.J.A. Sloane. “Sphere Packings, Lattices and Groups”. Springer-Verlag (1998).
  • [5]
    A. de Andrade & R. Palazzo Jr. Linear codes over finite rings. TEMA - Trends in Applied and Computational Mathematics, 6(2) (2005), 207–217.
  • [6]
    B. Erez. The Galois structure of the trace form in extensions of odd prime degree. Journal of Algebra, 118(2) (1988), 438–446.
  • [7]
    J.C. Interlando, J.O.D. Lopes & T.P. Nóbrega. The discriminant of abelian number fields. Journal of Algebra and its Applications, 5(01) (2006), 35–41.
  • [8]
    F. Oggier, E. Bayer-Fluckiger & E. Viterbo. New algebraic constructions of rotated cubic lattice constellations for the Rayleigh fading channel. In “Proceedings 2003 IEEE Information TheoryWorkshop (Cat. No. 03EX674)”. IEEE (2003), pp. 263–266.
  • [9]
    P. Ribenboim. “Classical theory of algebraic numbers”. Springer Science & Business Media (2013).
  • [10]
    J.P.O. Santos. “Introduction to numbers theory, Projeto Euclides”. Impa (2006).
  • [11]
    J.P. Serre. “A course in arithmetic”, volume 7. Springer Science & Business Media (2012).

Publication Dates

  • Publication in this collection
    13 Dec 2019
  • Date of issue
    Sep-Dec 2019

History

  • Received
    14 Mar 2019
  • Accepted
    6 Aug 2019
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