ISSN 2148-838Xhttps://doi.org/10.13069/jacodesmath.1000778 J. Algebra Comb. Discrete Appl. 8(3) • 151–160 Received: 15 May 2020 Accepted: 15 May 2021 Journal of Algebra Combinatorics Discrete Structures and Applications Rotated Dn-lattices in dimensions power of 3∗ Research Article Agnaldo J. Ferrari, Grasiele C. Jorge, Antonio A. de Andrade Abstract: In this work, we present constructions of families of rotated Dn-lattices which may be good for signal transmission over both Gaussian and Rayleigh fading channels. The lattices are obtained as sublattices of a family of rotated Z ⊕Ak2 lattices, where Ak2 is a direct sum of k = 3 r−1−1 2 copies of the A2-lattice, using free Z-modules in Z[ζ3r + ζ−13r ]. 2010 MSC: 11H06, 11R18, 94B12 Keywords: Lattices, Cyclotomic fields, Signal transmission 1. Introduction A lattice Λ ⊆ Rn is a discrete set generated by integer combinations of n linearly independent vectors in Rn over R. Its packing density ∆(Λ) is the proportion of the space Rn covered by congruent disjoint spheres of maximum radius [8]. A lattice Λ has diversity m ≤ n if m is the maximum number such that for all y = (y1, . . . ,yn) ∈ Λ, with y 6= 0, there are at least m non-zero coordinates. Given a full diversity lattice Λ ⊆ Rn, with m = n, the minimum product distance is defined as dmin(Λ) = inf{ ∏n i=1 |yi| for all y = (y1, . . . ,yn) ∈ Λ, with y 6= 0} [5]. Lattices have been considered in different areas, especially in coding theory, and they have been studied in several papers, from different points of view [1–7, 9, 10, 12, 13, 15]. Signal constellations having lattice structure have been studied for signal transmission over both Gaussian and single-antenna Rayleigh fading channel [7]. Usually the problem of finding good signal constellations for a Gaussian ∗ This work was supported by CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) under Grants No. 432735/2016-0 and 429346/2018-2 and Fapesp (Fundação de Amparo à Pesquisa do Estado de São Paulo) under Grant No. 2013/25977-7. Agnaldo J. Ferrari (Corresponding Author); Department of Mathematics, São Paulo State University, Bauru, SP 17033-360, Brazil (email: agnaldo.ferrari@unesp.br). Grasiele C. Jorge; Institute of Science and Technology, Federal University of São Paulo, São José dos Campos, SP 12247-014, Brazil (email: grasiele.jorge@unifesp.br). Antonio A. de Andrade; Department of Mathematics, São Paulo State University, São José do Rio Preto, SP 15054-000, Brazil (email: antonio.andrade@unesp.br). 151 https://orcid.org/0000-0002-1422-1416 https://orcid.org/0000-0002-1474-6001 https://orcid.org/0000-0001-6452-2236 A. J. Ferrari et. al. / J. Algebra Comb. Discrete Appl. 8(3) (2021) 151–160 channel is associated to the search for lattices with high packing density [8]. On the other hand, for a Rayleigh fading channel the efficiency is strongly related to the lattice diversity and minimum product distance [5, 7]. The approach in this work, following [12] and [13] is the use of algebraic number theory to construct rotated Dn-lattices with full diversity via free Z-modules. In [1, 4, 5] some families of rotated Zn-lattices for n = p−1 2 , where p ≥ 5 is a prime number, and n = 2s, for s ≥ 1, with full diversity and good minimum product distance are studied for transmission over Rayleigh fading channels. In [12, 13] are studied some families of rotated Dn-lattices with full diversity and good minimum product distance for transmission over both Gaussian and Rayleigh fading channels. In [12] are constructed rotated Dn-lattices for n = (p − 1)/2, where p ≥ 7 is a prime and n = 2k, for k ≥ 2 integer, and in [13] families of rotated Dn-lattices for n = 2k(p− 1), with k ≥ 0 integer and p ≥ 5 a prime, and n = (p− 1)(q − 1)/4, where p,q ≥ 5 are distinct prime numbers. In this work, we construct families of rotated Dn-lattices with full diversity n for n = 3s, s ≥ 1, (Propositions 3.4 and 3.5). A Dn-lattice has better packing density δ(Dn) when compared to Zn, i.e., Dn has the best lattice packing density for n = 3, 4, 5 and limn−→∞ δ(Zn) δ(Dn) = 0, and also a very efficient decoding algorithm [8]. 2. Algebraic lattices Let {v1, . . . ,vm} be a set of linearly independent vectors in Rn and Λ = { ∑m i=1 aivi; ai ∈ Z} the associated lattice. The set {v1, . . . ,vm} is called a basis for Λ. A matrix M whose rows are these vectors is said to be a generator matrix for Λ while the associated Gram matrix is G = MMt = (〈vi,vj〉) m i,j=1 . The determinant of Λ is det Λ = det G and it is an invariant under change of basis (see [8, p. 4]). Two lattices Λ1 and Λ2 are said to be similar if there is an orthogonal mapping φ : Rn → Rn and a real positive number c such that cφ(Λ1) = Λ2. When c = 1 the similar lattices Λ1 and Λ2 are said to be congruent or isomorphic. In this paper, as in [5, 12], we will say that Λ1 is a rotated Λ2-lattice if Λ1 and Λ2 are congruent. Let K be a totally real number field of degree n and OK its ring of integes. Let σi, for i = 1, . . . ,n, be the n distinct Q-homomorphisms from K to R. The canonical embedding σ : K −→ Rn is defined by σ(x) = (σ1(x), . . . ,σn(x)) [14, 16]. It can be shown that if I ⊆ OK is a free Z-module of rank n with Z-basis {w1, . . . ,wn}, then the image Λ = σ(I) is a lattice in Rn with basis {σ(w1), . . . ,σ(wn)} [16, Chapter 8] and it has full diversity [2, 5]. A Gram matrix for σ(I) is G = ( TrK|Q(wiwj) )n i,j=1 , where TrK|Q(x) = ∑n i=1 σi(x) for any x ∈ K [5]. In what follows let q(ui,uj) = TrK|Q(uiuj) for any ui,uj ∈ K. In this paper, we focus on the maximal totally real subfields of the cyclotomic fields Q(ζ3r ), where ζ3r is a primitive 3r-th root of unity, with r ≥ 3 a positive integer [17]. 3. Rotated Dn-lattices via K = Q(ζ3r + ζ−13r ), where r ≥ 3 and n = 3r−1 In [13, Proposition 2.7] it was shown that if K is a totally real Galois extension with dK an odd integer, then it is impossible to construct rotated Dn-lattices via fractional ideals of OK. In particular, it is impossible to construct rotated Dn-lattices via fractional ideals of Z[ζ3r +ζ−13r ] since dK = 3 2(r+1)3r−1−3r−1 2 by [11]. Thus, in this section, we present some families of rotated Dn-lattices using free Z-modules in Z[ζ3r +ζ−13r ]. Our strategy is to construct these lattices as sublattices of a family of rotated Z⊕A k 2-lattices, where Ak2 is a direct sum of k = 3r−1−1 2 copies of the A2-lattice. In [3] is presented a family of rotated Z ⊕Ak2-lattices as the image of a twisted embedding [2] applied to Z[ζ3r + ζ −1 3r ]. In Proposition 3.3, we construct a family of rotated Z ⊕Ak2-lattices using the canonical embedding, where the Lemma 3.1 and Proposition 3.2 are support for the proof of Proposition 3.3. 152 A. J. Ferrari et. al. / J. Algebra Comb. Discrete Appl. 8(3) (2021) 151–160 Lemma 3.1. [9] Consider e0 = 1 and ei = ζi3r + ζ −i 3r , for i = 1, 2, . . . , 3 r−1 − 1. 1. If i = 0, . . . , 3r−1 − 1, then q(ei,ei) = { 3r−1 if i = 0, 2 · 3r−1 otherwise. 2. If i = 1, 2, . . . , 3r−1 − 1, then q(ei,e0) = 0. 3. If i,j = 1, . . . , 3r−1 − 1, with i 6= j, then q(ei,ej) = { −3r−1 if i + j = 3r−1, 0 otherwise. Proposition 3.2. Consider u0 = e0, u1 = e1 and for i = 2, 3, . . . , 3r−1 − 1 ui = { ei+1 2 if i ≡ 1 (mod 2), e3r−1− i 2 otherwise. 1. If i = 0, . . . , 3r−1 − 1, then q(ui,ui) = { 3r−1 if i = 0, 2 · 3r−1 otherwise. 2. If i = 1, 2, . . . , 3r−1 − 1, then q(ui,u0) = 0. 3. If i,j = 1, . . . , 3r−1 − 1, with i 6= j, then q(ui,uj) = { −3r−1 if i + j ≡ 3 (mod 4) and |i− j| = 1, 0 otherwise. Proof. From Lemma 3.1, it follows that q(u0,u0) = q(e0,e0) = 3r−1 and for i = 1, 2, . . . , 3r−1 − 1, it follows that q(ui,ui) = 2 · 3r−1 and q(ui,u0) = q(ui,e0) = 0, for ui ∈ {e1,e2, . . . ,e3r−1−1}. If i,j = 1, 2, . . . , 3r−1 − 1, with i 6= j, then q(ui,uj) =   q(ei+1 2 ,ej+1 2 ) if i,j ≡ 1 (mod 2), q(ei+1 2 ,e3r−1−j 2 ) if i ≡ 1 and j ≡ 0 (mod 2), q(e3r−1− i 2 ,ej+1 2 ) if i ≡ 0 and j ≡ 1 (mod 2), q(e3r−1− i 2 ,e3r−1−j 2 ) if i,j ≡ 0 (mod 2). For i,j ≡ 1 (mod 2), it follows that either i + j ≡ 0 (mod 4) or i + j ≡ 2 (mod 4) and i+1 2 + j+1 2 6= 3r−1. Otherwise, since i 6= j, it follows that i = j = 3r−1 − 1, which is a contradiction. Thus, q(ui,uj) = 0. For i ≡ 1 (mod 2) and j ≡ 0 (mod 2), it follows that i+1 2 + 3r−1 − j 2 = 3r−1 if and only if i = j − 1. For i ≡ 0 (mod 2) and j ≡ 1 (mod 2), it follows that 3r−1 − i 2 + j+1 2 = 3r−1 if and only if j = i− 1. In the last two cases, as i + j is odd, it follows that i + j ≡ 3 (mod 4), because if i + j ≡ 1 (mod 4), with i = j−1 (respectively, j = i−1), it follows that j is odd (respectively, i is odd), which is a contradiction. Therefore, q(ui,uj) = −3r−1 if i + j ≡ 3 (mod 4) and |i− j| = 1. For i,j ≡ 0 (mod 2), it follows that either i + j ≡ 0 (mod 4) or i + j ≡ 2 (mod 4) and 3r−1 − i 2 + 3r−1 − j 2 6= 3r−1. Otherwise, since i 6= j, it follows that i = j = 3r−1 − 1, which is a contradiction. Thus, q(ui,uj) = 0. Proposition 3.3. The lattice 1 √ 3r−1 σ(OK) is a rotated version of Z⊕Ak2, where Ak2 is a direct sum of k = 3 r−1−1 2 copies of the A2-lattice. Proof. From Proposition 3.2, it follows that {u0,u1, . . . ,u3r−1−1} is a Z-basis of OK because it is a permutation of the Z-basis {e0,e1, . . . ,e3r−1−1}. A generator matrix of the algebraic lattice 1√3r−1 σα(OK) 153 A. J. Ferrari et. al. / J. Algebra Comb. Discrete Appl. 8(3) (2021) 151–160 is given by M = 1√ 3r−1 N, where N = (σi(uj−1))3 r−1 i,j=1, and the associated Gram matrix is given by G = MMt = 1 3r−1 (q(ui,uj)) 3r−1−1 i,j=0 . So, G =   1 2 −1 −1 2 2 −1 −1 2 ... 2 −1 −1 2   . It follows that the matrix G is a Gram matrix of Z⊕Ak2-lattice. In what follows, we split in two cases, i.e., we construct rotated Dn-lattices for n = 3r−1, for r even and for r odd. 3.1. Rotated Dn-lattices for n = 3r−1, where r ≥ 4 is even In this section, we present a construction of rotated Dn-lattices using Z-modules in the totally real number field K = Q(ζ3r + ζ−13r ), where r is even. The Dn-lattice is obtained as sublattice of Z⊕A k 2 using B = {u0,u1, . . . ,u3r−1−1} a Z-basis of OK. Proposition 3.4. Let I = Zω0 ⊕Zω1 ⊕ . . .⊕Zω3r−1−1 be a free Z-module of OK, where 1. ω0 = −4u0 − 2u1 − 2u2; ω1 = −2u1 + 2u2; ω2 = 4u0 − 2u2; ω3 = −2u0 + 2u1 + 2u2 −u5 + u6 −u9 + u10; 2. For j = 1, 2 . . . , 3 r−1−11 8 , ω4j = u8j−3 −u8j−2 −u8j−1 + u8j + u8j+1 −u8j+2 −u8j+3 + u8j+4; ω4j+1 = −u8j−3 + u8j−2 + u8j−1 −u8j + u8j+1 −u8j+2 + u8j+3 −u8j+4; ω4j+2 = u8j−3 −u8j−2 −u8j−1 + u8j −u8j+1 + u8j+2 + u8j+3 −u8j+4; ω4j+3 = u8j−1 −u8j −u8j+3 + u8j+4 −u8j+5 + u8j+6 −u8j+9 + u8j+10; ω3r−1+1−4j = −u8j−3 −u8j−2 −u8j−1 −u8j + 3u8j+1 + 3u8j+2 −u8j+3 −u8j+4; ω3r−1+2−4j = −u8j−3 −u8j−2 + 3u8j−1 + 3u8j −u8j+1 −u8j+2 + u8j+3 + u8j+4; ω3r−1+3−4j = 3u8j−3 + 3u8j−2 −u8j−1 −u8j + u8j+1 + u8j+2 + u8j+3 + u8j+4; If j 6= 1, ω3r−1+4−4j = −u8j−9 −u8j−8 − 2u8j−7 − 2u8j−6 + u8j−5 + u8j−4 −u8j−3 −u8j−2 −u8j+1 −u8j+2 − 2u8j+3 − 2u8j+4; 3. For j = 3 r−1−3 8 , ω4j = u3 −u4 + u8j−3 −u8j−2 −u8j−1 + u8j + u8j+1 −u8j+2; ω4j+1 = −u3 + u4 −u8j−3 + u8j−2 + u8j−1 −u8j + u8j+1 −u8j+2; ω4j+2 = −u3 + u4 + u8j−3 −u8j−2 −u8j−1 + u8j −u8j+1 + u8j+2; ω4j+3 = 2u3 − 2u8j − 2u8j+1 − 2u8j+2; ω3r−1+1−4j = −u3 −u4 −u8j−3 −u8j−2 −u8j−1 −u8j + 3u8j+1 + 3u8j+2; ω3r−1+2−4j = u3 + u4 −u8j−3 −u8j−2 + 3u8j−1 + 3u8j −u8j+1 154 A. J. Ferrari et. al. / J. Algebra Comb. Discrete Appl. 8(3) (2021) 151–160 −u8j+2; ω3r−1+3−4j = u3 + u4 + 3u8j−3 + 3u8j−2 −u8j−1 −u8j + u8j+1 + u8j+2; ω3r−1+4−4j = −2u3 − 2u4 + u8j−9 + u8j−8 − 2u8j−7 − 2u8j−6 + u8j−5 + u8j−4 −u8j−3 −u8j−2 −u8j+1 −u8j+2. Therefore, Λ = 1 2 √ 3r σ(I) ⊆ R3 r−1 is a rotated version of the D3r−1 -lattice. Proof. From Proposition 3.2, it follows that q(ω0,ω0) = TrK/Q(ω0ω0) = TrK/Q((−4u0 − 2u1 − 2u2)(−4u0 − 2u1 − 2u2) = TrK/Q(16u0u0 + 16u0u1 + 16u0u2 + 4u1u1 + 8u1u2 + 4u2u2) = 16q(u0,u0) + 16q(u0,u1) + 16q(u0,u2) + 4q(u1,u1) + 8q(u1,u2) + 4q(u2,u2) = 24 · 3r−1. q(ω1,ω1) = 4q(u1,u1) + 4q(u2,u2) − 8q(u1,u2) = 24 · 3r−1. q(ω2,ω2) = 16q(u0,u0) + 4q(u2,u2) = 24 · 3r−1. q(ω3,ω3) = 4q(u0,u0) + 4q(u1,u1) + 8q(u1,u2) + 4q(u2,u2) +q(u5,u5) − 2q(u5,u6) + q(u6,u6) + q(u9,u9) −2q(u9,u10) + q(u10,u10) = 24 · 3r−1. q(ω0,ω1) = q(ω0,ω3) = q(ω1,ω3) = 0. q(ω0,ω2) = q(ω1,ω2) = q(ω2,ω3) = q(ω3,ω4) = −12 · 3r−1. Let j = 1, 2, . . . , 3 r−1−3 8 . Since q(ui,uj) 6= 0 if and only if i + j ≡ 3 (mod 4) and |i−j| = 1, it follows that q(ω4j,ω4j) = q(u8j−3,u8j−3) − 2q(u8j−3,u8j−2) + q(u8j−2,u8j−2) + q(u8j−1,u8j−1) − 2q(u8j−1,u8j) + q(u8j,u8j) + q(u8j+1,u8j+1) − 2q(u8j+1,u8j+2) + q(u8j+2,u8j+2) + q(u8j+3,u8j+3) − 2q(u8j+3,u8j+4) + q(u8j+4,u8j+4) = 24 · 3r−1. Similarly, q(ω3r−1+1−4j,ω3r−1+1−4j = q(u8j−3,u8j−3) + 2q(u8j−3,u8j−2) + q(u8j−2,u8j−2) + q(u8j−1,u8j−1) + 2q(u8j−1,u8j) + q(u8j,u8j) + 9q(u8j+1,u8j+1) + 18q(u8j+1,u8j+2) + 9q(u8j+2,u8j+2) + q(u8j+3,u8j+3) + 2q(u8j+3,u8j+4) + q(u8j+4,u8j+4) = 24 · 3r−1, q(ω4j+1,ω4j+1) = q(ω4j+2,ω4j+2) = q(ω4j+3,ω4j+3) = = q(ω3r−1+2−4j,ω3r−1+2−4j) = q(ω3r−1+3−4j,ω3r−1+3−4j) = q(ω3r−1+4−4j,ω3r−1+4−4j) = 24 · 3r−1, q(ω4j,ω4j+1) = −q(ω8j−3,ω8j−3) + 2q(ω8j−3,ω8j−2) −q(ω8j−2,ω8j−2) −q(ω8j−1,ω8j−1) + 2q(ω8j−1,ω8j) −q(ω8j,ω8j) + q(ω8j+1,ω8j+1) − 2q(ω8j+1,ω8j+2) + q(ω8j+2,ω8j+2) −q(ω8j+3,ω8j+3) + 2q(ω8j+3,ω8j+4) −q(ω8j+4,ω8j+4) = −12 · 3r−1, q(ω4j+1,ω4j+2) = q(ω4j+2,ω4j+3) = q(ω4j+3,ω4(j+1)) = = q(ω3r−1+1−4j,ω3r−1+2−4j) = q(ω3r−1+2−4j,ω3r−1+3−4j) = q(ω3r−1+3−4j,ω3r−1+4−4j) = −12 · 3r−1. Finally, for k,l = 1, 2, . . . , 3r−1 − 2, with l > k + 1, it follows that q(ωk,ωl) = 0. Now, C = {ω0,ω1, . . . ,ω3r−1−1} is a basis of a free Z-module I. A generator matrix of the algebraic lattice 1 2 √ 3r σ(I) is given by M = 1 2 √ 3r N, where N = (σi(ωj−1))3 r−1 i,j=1, and the associated Gram matrix is 155 A. J. Ferrari et. al. / J. Algebra Comb. Discrete Appl. 8(3) (2021) 151–160 G = MMt = 1 12 · 3r−1 (q(ωi,ωj)) 3r−1−1 i,j=0 = =   2 0 −1 0 0 0 . . . 0 0 2 −1 0 0 0 . . . 0 −1 −1 2 −1 0 0 . . . 0 0 0 −1 2 −1 0 . . . 0 0 0 0 −1 2 −1 . . . 0 ... ... ... ... ... ... ... ... 0 0 0 . . . 0 −1 2 −1 0 0 0 . . . 0 0 −1 2   . Therefore, G is the Gram matrix of a D3r−1-lattice. 3.2. Rotated Dn-lattices for n = 3r−1, where r ≥ 3 is odd In this section, we present a construction of rotated Dn-lattices using Z-modules via the totally real number field K = Q(ζ3r + ζ−13r ), where r is odd. The Dn-lattice is obtained as sublattice of Z⊕A k 2 using B = {u0,u1, . . . ,u3r−1−1} a Z-basis of OK. Proposition 3.5. Let I = Zω0 ⊕Zω1 ⊕ . . .⊕Zω3r−1−1 be a free Z-module of OK, where 1. ω0 = −6u0 − 3u1 − 3u3; ω1 = 6u0 − 3u1 − 3u3; ω2 = 6u1; 2. For 3 ≤ j ≤ 3 r−1−3 2 , where j is odd, ωj = −3u2j−5 + 3u2j−3 − 3u2j−1 − 3u2j+1; ωj+1 = 6u2j−1; 3. For j = 3 r−1+1 2 , ωj = −u2j−7 − 2u2j−6 − 5u2j−5 − 4u2j−4 + 4u2j−3 + 2u2j−2; ωj+1 = −u2j−9 − 2u2j−8 −u2j−7 − 2u2j−6 + 3u2j−5 + 6u2j−4 −u2j−3 − 2u2j−2; ωj+2 = −u2j−9 − 2u2j−8 + 3u2j−7 + 6u2j−6 −u2j−5 − 2u2j−4 + u2j−3 + 2u2j−2; ωj+3 = 3u2j−9 + 6u2j−8 −u2j−7 − 2u2j−6 + u2j−5 + 2u2j−4 + u2j−3 + 2u2j−2; 4. For j = 1, 2, . . . , 3 r−1−9 8 , with r > 3, ω3r−1−4j = −u8j−5 − 2u8j−4 − 2u8j−3 − 4u8j−2 + u8j−1 + 2u8j −u8j+1 − 2u8j+2 −u8j+5 − 2u8j+6 − 2u8j+7 − 4u8j+8; ω3r−1+1−4j = −u8j−7 − 2u8j−6 −u8j−5 − 2u8j−4 + 3u8j−3 + 6u8j−2 −u8j−1 − 2u8j; ω3r−1+2−4j = −u8j−7 − 2u8j−6 + 3u8j−5 + 6u8j−4 −u8j−3 − 2u8j−2 + u8j−1 + 2u8j; ω3r−1+3−4j = 3u8j−7 + 6u8j−6 −u8j−5 − 2u8j−4 + u8j−3 + 2u8j−2 + u8j−1 + 2u8j. Therefore, Λ = 1 6 √ 3r−1 σ(I) ⊆ R3 r−1 is a rotated version of a D3r−1 -lattice. Proof. From Proposition 3.2, it follows that q(ω0,ω0) = TrK/Q(ω0ω0) = TrK/Q((−6u0 − 3u1 − 3u3)(−6u0 − 3u1 − 3u3) = TrK/Q(36u0u0 + 36u0u1 + 36u0u3 + 9u1u1 + 18u1u3 + 9u3u3) = 36q(u0,u0) + 36q(u0,u1) 156 A. J. Ferrari et. al. / J. Algebra Comb. Discrete Appl. 8(3) (2021) 151–160 + 36q(u0,u3) + 9q(u1,u1) + 18q(u1,u3) + 9q(u3,u3) = 72 · 3r−1. q(ω1,ω1) = 36q(u0,u0) + 9q(u1,u1) + 9q(u3,u3) = 72 · 3r−1. q(ω2,ω2) = 36q(u1,u1) = 72 · 3r−1. q(ω0,ω1) = q(ω0,ω3) = q(ω1,ω3) = 0. q(ω0,ω2) = q(ω1,ω2) = q(ω2,ω3) = −36 · 3r−1. Let 3 ≤ j ≤ 3 r−1−3 2 , with j odd. Since q(ui,uj) 6= 0 if and only if i + j ≡ 3 (mod 4) and |i− j| = 1, it follows that q(ωj,ωj) = 9q(u2j−5,u2j−5) + 9q(u2j−3,u2j−3) + 9q(u2j−1,u2j−1) + 9q(u2j+1,u2j+1) = 72 · 3r−1. q(ωj+1,ωj+1) = 36q(u2j−1,u2j−1) = 72 · 3r−1. Furthermore, q(ωj,ωj+1) = −18q(u2j−1,u2j−1) = −36 · 3r−1, and for j < 3 r−1−3 2 , q(ωj+1,ωj+2) = q(6u2j−1,−3u2(j+2)−5 + 3u2(j+2)−3 − 3u2(j+2)−1 − 3u2(j+2)+1) = q(6u2j−1,−3u2j−1 + 3u2j+1 − 3u2j+3 − 3u2j+5) = −18q(u2j−1,u2j−1) = = −36 · 3r−1. For j = 3 r−1+1 2 , it follows that q(ωj,ωj) = q(u2j−7,u2j−7) + 4q(u2j−7,u2j−6) + 4q(u2j−6,u2j−6) + 25q(u2j−5,u2j−5) + 40q(u2j−5,u2j−4) + 16q(u2j−4,u2j−4) + 16q(u2j−3,u2j−3) + 16q(u2j−3,u2j−2) + 4q(u2j−2,u2j−2) = 72 · 3r−1. In the same way, it follows that q(ωj+1,ωj+1) = q(ωj+2,ωj+2) = q(ωj+3,ωj+3) = 72 · 3r−1. Also, q(ωj,ωj+1) = q(ω2j−7,ω2j−7) + 4q(ω2j−7,ω2j−6) + 4q(ω2j−6,ω2j−6) − 15q(ω2j−5,ω2j−5) − 42q(ω2j−5,ω2j−4) − 24q(ω2j−4,ω2j−4) − 4q(ω2j−3,ω2j−3) − 10q(ω2j−3,ω2j−2) − 4q(ω2j−2,ω2j−2) = −36 · 3r−1. In the same way, it follows that q(ωj+1,ωj+2) = q(ωj+2,ωj+3) = −36 · 3r−1. and for k = 3 r−1−9 8 , q(ωj+3,ω3r−1−4k) = q(ω3r−1+7 2 ,ω3r−1+9 2 ) = −36 · 3r−1. For j = 1, 2, . . . , 3 r−1−9 8 , with r > 3, q(ω3r−1−4j,ω3r−1−4j) = q(ω8j−5,ω8j−5) + 4q(ω8j−5,ω8j−4) + 4q(ω8j−4,ω8j−4) + 4q(ω8j−3,ω8j−3) + 16q(ω8j−3,ω8j−2) + 16q(ω8j−2,ω8j−2) + q(ω8j−1,ω8j−1) + 4q(ω8j−1,ω8j) + 4q(ω8j,ω8j) + q(ω8j+1,ω8j+1) + 4q(ω8j+1,ω8j+2) + 4q(ω8j+2,ω8j+2) + q(ω8j+5,ω8j+5) + 4q(ω8j+5,ω8j+6) + 4q(ω8j+6,ω8j+6) + 4q(ω8j+7,ω8j+7) + 16q(ω8j+7,ω8j+8) + 16q(ω8j+8,ω8j+8) = 72 · 3r−1. In the same way, it follows that q(ω3r−1+1−4j,ω3r−1+1−4j) = q(ω3r−1+2−4j,ω3r−1+2−4j) = q(ω3r−1+3−4j,ω3r−1+3−4j) = 72 · 3r−1. Also, q(ω3r−1−4j,ω3r−1+1−4j) = q(u8j−5,u8j−5) + 4q(u8j−5,u8j−4) + 4q(u8j−4,u8j−4) − 6q(u8j−3,u8j−3) − 24q(u8j−3,u8j−2) − 24q(u8j−2,u8j−2) −q(u8j−1,u8j−1) − 4q(u8j−1,u8j) − 4q(u8j,u8j) = −36 · 3r−1. In the same way, it follows that q(ω3r−1+1−4j,ω3r−1+2−4j) = q(ω3r−1+2−4j,ω3r−1+3−4j) = −36 · 3r−1. Finally, for k,l = 1, 2, . . . , 3r−1 − 2, with l > k + 1, it follows that q(ωk,ωl) = 0. Now, n C = {ω0,ω1, . . . ,ω3r−1−1} is a basis of a free Z-module I. A generator matrix of the algebraic 157 A. J. Ferrari et. al. / J. Algebra Comb. Discrete Appl. 8(3) (2021) 151–160 lattice 1 6 √ 3r−1 σ(I) is given by M = 1 6 √ 3r−1 N, where N = (σi(ωj−1))3 r−1 i,j=1, and the associated Gram matrix is G = MMt = 1 36 · 3r−1 (q(ωi,ωj)) 3r−1−1 i,j=0 = =   2 0 −1 0 0 0 . . . 0 0 2 −1 0 0 0 . . . 0 −1 −1 2 −1 0 0 . . . 0 0 0 −1 2 −1 0 . . . 0 0 0 0 −1 2 −1 . . . 0 ... ... ... ... ... ... ... ... 0 0 0 . . . 0 −1 2 −1 0 0 0 . . . 0 0 −1 2   . Therefore, G is the Gram matrix of a D3r−1-lattice. 4. Conclusions In this paper, we construct full diversity rotated versions of D3r−1-lattices via the canonical embed- ding and two families of Z-modules of the ring of the integers Z[ζ3r + ζ−13r ], for r ≥ 3 a positive integer, since it is impossible to construct rotated Dn-lattices via fractional ideals of Z[ζ3r + ζ−13r ] [13]. The lat- tices obtained here are sublattices of the family of rotated Z ⊕Ak2-lattices, where Ak2 is a direct sum of k = 3 r−1−1 2 copies of the A2-lattice. In [1] and [4] families of rotated Z2 r−2 -lattices were obtained via the ring of integers Z[ζ2r + ζ−12r ]. In [5] a family of rotated Z(p−1)/2-lattices was obtained via the ring of integers Z[ζp +ζ−1p ], with p prime. In [9] two families of rotated Z3 r−1 -lattices were obtained via free Z-modules of Z[ζ3r + ζ−13r ], one for r odd and one for r even. In [12] two families of rotated D2r−2-lattices were obtained, one via the ring of integers Z[ζ2r +ζ−12r ] and one via a principal ideal of Z[ζ2r +ζ −1 2r ]. Also in [12] a family of rotated D(p−1)/2-lattices was presented via free Z-modules in Z[ζp +ζ−1p ], with p prime, that are not ideals. In [13] considering the compositum of Q(ζ2r +ζ−12r ) and Q(ζp +ζ −1 p ) and the compositum of Q(ζp1 +ζ −1 p1 ) and Q(ζp2 +ζ −1 p2 ), where p,p1 and p2 are prime numbers with p1 6= p2, were constructed families of rotated Dn-lattices via free Z- modules of rank n that are not ideals. In Table 1, we list the number fields considered in [1, 4, 5, 9, 12, 13] and here for constructing rotated Zn and Dn-lattices for some values of n. Let K1 = Q(ζ2r + ζ−12r ), K2 = Q(ζp + ζ−1p ), where p is a prime, K3 = Q(ζ2r + ζ −1 2r )Q(ζp + ζ −1 p ), K4 = Q(ζp1 + ζ −1 p1 )Q(ζp2 + ζ −1 p2 ), with p1 6= p2, and K5 = Q(ζ3r + ζ−13r ). We observe that for r = 14, 21, 25, 26, 28, 29 and 30 there are not p,p1,p2 prime numbers with p1 6= p2 such that the degree of Q(ζp + ζ−1p ) and Q(ζp1 + ζ−1p1 )Q(ζp2 + ζ −1 p2 ) be 3r−2. n Zn Dn K1 K2 K5 K1 K2 K3 K4 K5 2 r = 3 p = 5 − − − − − − 3 − p = 7 r = 3 − p = 7 − − r = 3 4 r = 4 − − r = 4 − r = 3, p = 5 − − 8 r = 5 p = 17 − r = 5 p = 17 r = 4, p = 5 − − 9 − p = 19 r = 4 − p = 19 − − r = 4 16 r = 6 − − r = 6 − r = p = 5 p1,p2 ∈{5,17} − 27 − − r = 5 − − − p1,p2 ∈{7,19} r = 5 32 r = 7 − − r = 7 − r = 4, p = 17 − − 64 r = 8 − − r = 8 − r = 7, p = 5 − − 81 − p = 163 r = 6 − p = 163 − − r = 6 158 A. J. Ferrari et. al. / J. Algebra Comb. Discrete Appl. 8(3) (2021) 151–160 n Zn Dn K1 K2 K5 K1 K2 K3 K4 K5 128 r = 9 p = 257 − r = 9 p = 257 r = 8, p = 5 − − 243 − p = 487 r = 7 − p = 487 − p1,p2 ∈{7,163} r = 7 256 r = 10 − − r = 10 − r = 7, p = 17 p1,p2 ∈{5,257} − 512 r = 11 − − r = 11 − r = 10, p = 5 − − 729 − p = 1459 r = 8 − p = 1459 − p1,p2 ∈{7,487} r = 8 p1,p2 ∈{19,163} 1024 r = 12 − − r = 12 − r = 9, p = 17 p1,p2 ∈{17,257} − 2048 r = 13 − − r = 13 − r = 10, p = 17 − − 2187 − − r = 9 − − − p1,p2 ∈{7,1459} r = 9 p1,p2 ∈{19,487} 4096 r = 14 − − r = 14 − r = 13, p = 5 − − 6561 − − r = 10 − − − p1,p2 ∈{19,1459} r = 10 8192 r = 15 − − r = 15 − r = 12, p = 17 − − 16384 r = 16 − − r = 16 − r = 13, p = 5 − − 19683 − p = 39367 r = 11 − p = 39367 − p1,p2 ∈{163,487} r = 11 32768 r = 17 p = 65537 − r = 17 p = 65537 r = 14, p = 17 − − 59049 − − r = 12 − − − p1,p2 ∈{7,39367} r = 12 p1,p2 ∈{163,1459} 65536 r = 18 − − r = 18 − r = 17, p = 5 − − 131072 r = 19 − − r = 19 − r = 16, p = 17 − − 177147 − − r = 13 − − − p1,p2 ∈{19,39367} r = 13 p1,p2 ∈{487,1459} 262144 r = 20 − − r = 20 − r = 13, p = 257 − − 524288 r = 21 − − r = 21 − r = 20, p = 5 − − 531441 − − r = 14 − − − − r = 14 1048576 r = 22 − − r = 22 − r = 19, p = 17 − − 1594323 − − r = 15 − − − p1,p2 ∈{163,39367} r = 15 Table 2. 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Washington, Introduction to cyclotomic fields, Springer-Verlag, New York (1982). 160 https://doi.org/10.13069/jacodesmath.729440 https://doi.org/10.13069/jacodesmath.729440 https://doi.org/10.13069/jacodesmath.729440 https://doi.org/10.1142/S0219498806001636 https://doi.org/10.1142/S0219498806001636 https://doi.org/10.1016/j.jnt.2012.05.002 https://doi.org/10.1016/j.jnt.2012.05.002 https://doi.org/10.1007/s00013-013-0501-8 https://doi.org/10.1007/s00013-013-0501-8 https://doi.org/10.13069/jacodesmath.75353 https://doi.org/10.13069/jacodesmath.75353 https://doi.org/10.1007/978-1-4684-0133-2 Introduction Algebraic lattices Rotated Dn-lattices via K=Q(3r+3r-1), where r 3 and n=3r-1 Conclusions References