Rotated <inline-formula><mml:math display="inline"><mml:msup><mml:mstyle mathvariant="double-struck"><mml:mi>ℤ</mml:mi></mml:mstyle><mml:mi>n</mml:mi></mml:msup></mml:math></inline-formula>-Lattices via Real Subfields of <inline-formula><mml:math display="inline"><mml:mrow><mml:mstyle mathvariant="double-struck"><mml:mi>ℚ</mml:mi></mml:mstyle><mml:mo form="prefix" stretchy="false">(</mml:mo><mml:msub><mml:mi>ζ</mml:mi><mml:msup><mml:mn>2</mml:mn><mml:mi>r</mml:mi></mml:msup></mml:msub><mml:mo form="postfix" stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula>

ANDRADE, A. A.; INTERLANDO, J. C.

doi:10.5540/tema.2019.020.03.0445

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Articles • TEMA (São Carlos) 20 (3) • Sep-Dec 2019 • https://doi.org/10.5540/tema.2019.020.03.0445 copy

Rotated $ℤ^{n}$ -Lattices via Real Subfields of $ℚ (ζ_{2^{r}})$

Authorship SCIMAGO INSTITUTIONS RANKINGS

ABSTRACT

A method for constructing rotated $ℤ^{n}$ -lattices, with n a power of $2$ , based on totally real subfields of the cyclotomic field $ℚ (ζ_{2^{r}})$ , where $r \geq 4$ is an integer, is presented. Lattices exhibiting full diversity in some dimensions $n$ not previously addressed are obtained.

Key words
lattices; cyclotomic fields; modulation design; fading channels; minimum product distance

$n$	$r$	$k$	$\sqrt[n]{d_{p, min} (Λ)}$	Bound	$n$	$r$	$k$	$\sqrt[n]{d_{p, min} (Λ)}$	Bound
$2$	$3$	$0$	$0.594604$	$0.668740$	$2$	$4$	$1$	$0.594604$	$0.668740$
$4$	$4$	$0$	$0.385553$	$0.438993$	$4$	$5$	$1$	$0.385553$	0.438993
$8$	$5$	$0$	$0.261068$	$0.296367$	$8$	$6$	$1$	$0.261068$	$0.296367$
$16$	$6$	$0$	$0.180648$	$0.214279$	$16$	$7$	$1$	$0.180648$	$0.214279$
$32$	$7$	$0$	$0.126361$	$0.167500$	$32$	$8$	$1$	$0.126361$	$0.167500$
$64$	$8$	$0$	$0.088868$	$-$	$64$	$9$	$1$	$0.088868$	$-$
$128$	$9$	$0$	$0.062669$	$-$	$128$	$10$	$1$	$0.062669$	$-$
$256$	$10$	$0$	$0.044254$	$-$	$256$	$11$	$1$	$0.044254$	$-$
$512$	$11$	$0$	$0.031271$	$-$	$512$	$12$	$1$	$0.031271$	$-$
$1024$	$12$	$0$	$0.022105$	$-$	$1024$	$13$	$1$	$0.022105$	$-$
$2048$	$13$	$0$	$0.015628$	$-$	$2048$	$14$	$1$	$0.015628$	$-$

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[1] *Corresponding author: Antonio A. Andrade – E-mail: antonio.andrade@unesp.br – 0000-0001-6452-2236

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Rotated ℤn-Lattices via Real Subfields of ℚ(ζ2r)

ABSTRACT

Rotated $ℤ^{n}$ -Lattices via Real Subfields of $ℚ (ζ_{2^{r}})$