Constructions of Dense Lattices of Full Diversity

A lattice construction using ℤ-submodules of rings of integers of number fields is presented. The construction yields rotated versions of the laminated lattices $Λ n f o r n = 2, 3, 4, 5, 6$ , 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

Authorship

A. A. ANDRADE ^**Corresponding author: Antonio Aparecido de Andrade - E-mail: antonio.andrade@unesp.br

Departamento 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.brUniversidade Estadual Paulista “Júlio de Mesquita Filho”BrasilSP, BrasilDepartamento 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

http://orcid.org/0000-0001-6452-2236

A. J. FERRARI

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

http://orcid.org/0000-0002-1422-1416

J. C. INTERLANDO

Department of Mathemtaics & Statistics, San Diego State University, San Diego, California, USA - E-mail: interlan@sdu.eduSan Diego State UniversityUSASan Diego, California, USADepartment of Mathemtaics & Statistics, San Diego State University, San Diego, California, USA - E-mail: interlan@sdu.edu

http://orcid.org/0000-0003-4928-043X

R.R. ARAUJO

Instituto Federal de São Paulo, Cubatão, SP, Brasil - E-mail: dearaujorobisonricardo@gmail.comInstituto Federal de São PauloBrasilCubatão, SP, BrasilInstituto Federal de São Paulo, Cubatão, SP, Brasil - E-mail: dearaujorobisonricardo@gmail.com

http://orcid.org/0000-0002-1357-9926

*Corresponding author: Antonio Aparecido de Andrade - E-mail: antonio.andrade@unesp.br

SCIMAGO INSTITUTIONS RANKINGS

Formulas

Formulas (44)

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d_{p, m i n} (Λ) = m i n \{\prod_{i = 1}^{n} |x_{i}| : 0 \neq (x_{1}, . . ., x_{n}) \in Λ\} .

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))),

δ (Λ) = \frac{t^{n / 2}}{2^{n} \sqrt{|d_{K}|} [O_{K} : M]} = \frac{{(\sqrt{t})}^{n}}{2^{n} \sqrt{|d_{K}|} [O_{K} : M]},

t = c_{k} m i n \{T r_{K / ℚ} (x \bar{x}) : x \in M, x \neq 0\}

ζ_{n} = ζ_{n}^{a u + b v} = ζ_{n}^{a u} ζ_{n}^{b v} = ζ_{b}^{u} ζ_{a}^{v}

\begin{matrix} φ : G a l (ℚ (ζ_{n}) / ℚ (ζ_{a})) \to G a l (ℚ (ζ_{b}) / ℚ) \\ σ \mapsto σ_{ℚ (ζ_{b})} \end{matrix}

T r_{ℚ (ζ_{n}) / ℚ (ζ_{a})} (α) = T r_{ℚ (ζ_{b} / d) / ℚ} (α) .

T r_{ℚ (ζ_{n}) / ℚ} (ζ_{n}^{j}) = \frac{φ (n)}{φ (n / d)} T r_{ℚ (ζ_{n / d}) / ℚ} (ζ_{n / d}^{j / d}) .

T r_{ℚ (ζ_{n}) / ℚ} (ζ_{p i}^{j}) = \{\begin{cases} - 1 & i f i = 1, \\ 0 & i f i > 1 . \end{cases}

T r_{ℚ (ζ_{n}) / ℚ} (ζ_{n}^{j}) = \frac{φ (n)}{φ (n / d)} μ (n / d),

T r_{ℚ (ζ_{n}) / ℚ} (ζ_{n}^{i}) \neq 0 \Leftrightarrow d = (n / P) t_{j} a n d i = (n / P) j,

T r_{ℚ (ζ_{n}) / ℚ} (x \bar{x}) = \frac{n}{P} \{φ (P) \sum_{i = 0}^{m - 1} a_{i}^{2} + 2 \sum_{j = 1}^{φ (P) - 1} μ (\frac{P}{t_{j}}) φ (t_{j}) A_{\frac{n}{P} j}\},

T r_{K / ℚ (x^{2})} = m \sum_{i = 1}^{m / 2} a_{i}^{2} + \frac{n}{P} (\sum_{\begin{matrix} i = u \\ 2 | \frac{n}{P} i \end{matrix}}^{φ (P)} ρ (t_{i}) a_{\frac{n}{2 P} i}^{2} + 2 \sum_{i = 1}^{s} ρ (t_{i}) A_{\frac{n}{P} i} + 2 \sum_{i = v}^{φ (P) - 1} ρ (t_{i}) B_{\frac{n}{P} i}) .

x = a_{1} (ζ_{n} + ζ_{n}^{- 1}) + a_{2} (ζ_{n}^{2} + ζ_{n}^{- 2}) + ... + a_{\frac{m}{2}} (ζ_{n}^{\frac{m}{2}} + ζ_{n}^{- \frac{m}{2}})

\begin{matrix} x^{2} = (a_{1} ζ_{n} + a_{1} ζ_{n}^{- 1} + ... + a_{\frac{m}{2}} ζ_{n}^{- \frac{m}{2}}) (a_{1} ζ_{n} + a_{1} ζ_{n}^{- 1} + ... + a_{\frac{m}{2}} ζ_{n}^{- \frac{m}{2}}) \\ = {[(a_{1} ζ_{n} + a_{2} ζ_{n}^{2} + ... + a_{\frac{m}{2}} ζ_{n}^{\frac{m}{2}}) + (a_{1} ζ_{n}^{- 1} + a_{2} ζ_{n}^{- 2} + ... + a_{\frac{m}{2}} ζ_{n}^{- \frac{m}{2}})]}^{2} \\ = A^{2} + {\bar{A}}^{2} + 2 A \bar{A}, \end{matrix}

x^{2} = \sum_{j = 1}^{m / 2} a_{j}^{2} (ζ_{n}^{2 j} + ζ_{n}^{- 2 j}) + 2 (\sum_{j = 3}^{m - 1} B_{j} β_{j}) + 2 (\sum_{j = 1}^{m / 2} a_{j}^{2} + \sum_{j = 1}^{m / 2 - 1} A_{j} β_{j}),

T r_{ℚ (ζ_{n}) / ℚ} (x^{2}) = [ℚ (ζ_{n}) : K] T r_{K / ℚ} (x^{2}),

t = T r_{K / ℚ} (x^{2}) = \frac{1}{2} T r_{ℚ (ζ_{n}) / ℚ} (x^{2}) .

\begin{matrix} t = \frac{1}{2} [T r_{ℚ (ζ_{n}) / ℚ} (2 \sum_{j = 1}^{m / 2} a_{j}^{2} + 2 \sum_{j = 1}^{m / 2 - 1} A_{j} β_{j} \sum_{j = 1}^{m / 2} a_{j}^{2} (ζ_{n}^{2 j} + ζ_{n}^{- 2 j}) + 2 \sum_{j = 3}^{m - 1} B_{j} β_{j})] \\ = m \sum_{j = 1}^{m / 2} a_{j}^{2} + \sum_{j = 1}^{m / 2} a_{j}^{2} + T r_{ℚ (ζ_{n}) / ℚ} (ζ_{n}^{2 j}) \\ + 2 (\sum_{j = 1}^{m / 2 - 1} A_{j} T r_{ℚ (ζ_{n}) / ℚ} (ζ_{n}^{j}) + \sum_{j = 3}^{m - 1} B_{j} T r_{ℚ (ζ_{n}) / ℚ} (ζ_{n}^{j})) . \end{matrix}

\begin{matrix} \sum_{j = 1}^{m / 2} a_{j}^{2} T r_{ℚ (ζ_{n}) / ℚ} (ζ_{n}^{2 j}) = \frac{n}{P} \sum_{\begin{matrix} i = u \\ 2 | \frac{n}{P} i \end{matrix}}^{φ (P)} μ (\frac{P}{t_{i}}) φ (t_{i}) a_{\frac{n}{2 P} i}^{2}, \\ \sum_{j = 1}^{m / 2 - 1} A_{j} T r_{ℚ (ζ_{n}) / ℚ} (ζ_{n}^{j}) = \frac{n}{P} \sum_{i = 1}^{s} μ (\frac{P}{t_{i}}) φ (t_{i}) A_{\frac{n}{2 P} i}, \\ \sum_{j = 3}^{m - 1} B_{j} T r_{ℚ (ζ_{n}) / ℚ} (ζ_{n}^{j}) = \frac{n}{P} \sum_{i = v}^{φ (P) - 1} μ (\frac{P}{t_{i}}) φ (t_{i}) B_{\frac{n}{2 P} i}, \end{matrix}

T r_{K / ℚ} (x^{2}) = m \sum_{i = 1}^{m / 2} a_{i}^{2} + \frac{n}{P} (\sum_{\begin{matrix} i = u \\ 2 | \frac{n}{P} i \end{matrix}}^{φ (P)} ρ (t_{i}) a_{\frac{n}{2 P} i}^{2} + 2 \sum_{i = 1}^{s} ρ (t_{i}) A_{\frac{n}{2 P} i} + 2 \sum_{i = v}^{φ (P) - 1} ρ (t_{i}) B_{\frac{n}{2 P} i}),

T R_{K / ℚ} (x^{2}) = (p^{r}) \sum_{i = 1}^{\frac{φ (p^{r})}{2}} a_{i}^{2} - p^{r - 1} (\sum_{\begin{matrix} i = u \\ 2 | \frac{π}{p} i \end{matrix}}^{p - 1} a_{i \frac{p^{r - 1}}{2}}^{2} + 2 \sum_{i = 1}^{\frac{p - 3}{2}} A_{i p^{r - 1}} + 2 \sum_{i = v}^{p - 2} B_{i p^{r - 1}}),

T r_{K / ℚ} (x^{2}) = φ (2 p^{r}) \sum_{i = 1}^{\frac{φ (2 p^{r})}{2}} a_{i}^{2} - p^{r - 1} (\sum_{\begin{matrix} i = u \\ 2 | i \end{matrix}}^{p - 1} a_{i \frac{p^{r - 1}}{2}}^{2} - 2 U + 2 V),

t_{i} = \gcd (i, P) = \gcd (i,2 p) = \{\begin{matrix} 1 if i is odd, \\ 2 if i is even . \end{matrix}

ρ (t_{i}) = \{\begin{matrix} 1 if i is odd, \\ -1 if i is even . \end{matrix}

T r_{K / ℚ} (x^{2}) = φ (p q) \sum_{i = 1}^{\frac{φ (p q)}{2}} a_{i}^{2} + U + 2 V + 2 W,

\begin{matrix} s = ⌊\frac{φ (p q)}{2} - 1⌋, A_{j} = a_{1} a_{j + 1} + a_{2} a_{j + 2} + . . . + a_{\frac{φ (p q)}{2} - j} a_{\frac{φ (p q)}{2}}, \\ B_{j} = \sum_{\begin{matrix} k \geq 1 \\ k < j - k \leq \frac{φ (p q)}{2} \end{matrix}} a_{k} a_{j - k}, \\ U = - (p - 1) \sum_{\begin{matrix} i = 2 \\ 2 p | i \end{matrix}}^{φ (p q)} a_{\frac{i}{2}}^{2} - (q - 1) \sum_{\begin{matrix} i = 2 \\ 2 q | i \end{matrix}}^{φ (p q)} a_{\frac{i}{2}}^{2} + \sum_{\begin{matrix} i = 2 \\ 2 | i, g c d (i, p q) = 1 \end{matrix}}^{φ (p q)} a_{\frac{i}{2}}^{2}, \\ V = - (p - 1) \sum_{\begin{matrix} i = 1 \\ p | i \end{matrix}}^{s} A_{i} - (q - 1) \sum_{\begin{matrix} i = 1 \\ q | i \end{matrix}}^{s} A_{i} + \sum_{\begin{matrix} i = 1 \\ g c d (i, p q) = 1 \end{matrix}}^{s} A_{i}, \\ W = - (p - 1) \sum_{\begin{matrix} i = 3 \\ p | i \end{matrix}}^{φ (p q)} B_{i} - (q - 1) \sum_{\begin{matrix} i = 3 \\ q | i \end{matrix}}^{φ (p q) - 1} B_{i} + \sum_{\begin{matrix} i = 3 \\ g c d (i, p q) = 1 \end{matrix}}^{φ (p q) - 1} B_{i}, \end{matrix}

t_{j} =gcd(j,pq)= \{\begin{matrix} p if j is a multiple of p \\ q if j is a multiple of q \\ 1 otherwise, \end{matrix}

p (t_{i}) = \{\begin{matrix} - (p-1) if j is a multiple of p \\ - (q-1) if j is a multiple of q \\ 1 otherwise . \end{matrix}

\begin{matrix} T r K / Q (x^{2}) = p a_{0}^{2} + p p_{1} ... p_{r} (- 2 \sum_{1 \leq i \leq j \leq p - 1} a_{i} a_{j} + (p - 1) \sum_{i = 1}^{p - 1} a_{i}^{2}) \\ = p a_{0}^{2} + p p_{1} ... p_{r} (\sum_{i = 1}^{p - 1} a_{i}^{2} + \sum_{1 \leq i \leq j \leq p - 1} {(a_{i} - a_{j})}^{2}) . \end{matrix}

M = \{a_{1} (ζ_{12} + ζ_{12}^{- 1}) + a_{2} (ζ_{12}^{2} + ζ_{12}^{- 2}) \in O_{K} : a_{1} + a_{2} \equiv 0 (m o d 2)\},

T r_{K / ℚ} (α^{2}) = 8 (a_{1}^{2} - a_{1} a_{2} + a_{2}^{2}) .

(δ (σ (M)) = \frac{({\sqrt{2^{3}}}^{2})}{2^{4} \sqrt{3}} = \frac{1}{2 \sqrt{3}},

M = \{a_{1} (ζ_{9} + ζ_{9}^{- 1}) + a_{2} (ζ_{9}^{2} + ζ_{9}^{- 2}) + a_{3} (ζ_{9}^{3} + ζ_{9}^{- 3}) \in O_{K} : 4 a_{1} + 4 a_{2} + a_{3} \equiv 0 (m o d 6) a n d a_{3} \equiv 0 (m o d 2)\},

T r_{K}_{/ ℚ} (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})

δ (σ (M)) = \frac{{(\sqrt{2 \cdot 3^{2}})}^{3}}{2^{4} \cdot 3^{3}} = \frac{1}{4 \sqrt{2}},

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}) \in O_{K} : 4 a_{1} + 3 a_{2} + 2 a_{3} + a_{4} \equiv 0 (m o d 6)\},

T r_{K / ℚ} (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}) .

δ (σ (M)) = \frac{{(\sqrt{2^{3} \cdot 3})}^{4}}{2^{9} \cdot 3^{2}} = \frac{1}{8},

M = \{a_{0} + a_{1} σ (t) + a_{2} σ^{2} (t) + a_{3} σ^{3} (t) + a_{4} σ^{4} (t) \in O_{K} : a_{0} \equiv 0 (m o d 2), - a_{0} + a_{1} + a_{2} + a_{3} + a_{4} \equiv 0 (m o d 5)\} .

T r_{K / ℚ} (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}),

δ (σ (M)) = \frac{{(\sqrt{2 \cdot 5^{2}})}^{5}}{2^{6} \cdot 5^{5}} = \frac{1}{8 \sqrt{2}},

T r_{K / ℚ} (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} .

δ (M) = \frac{{(\sqrt{2^{3} \cdot 3^{2}})}^{6}}{2^{12} \cdot 3^{6} \sqrt{3}} = \frac{1}{8 \sqrt{3}},

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[1] *Corresponding author: Antonio Aparecido de Andrade - E-mail: antonio.andrade@unesp.br