Rotated $ A_n $-lattice codes of full diversity

Agnaldo José Ferrari; Tatiana Miguel Rodrigues de Souza

doi:10.3934/amc.2020118

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2022, Volume 16, Issue 3: 439-447. Doi: 10.3934/amc.2020118

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Rotated $ A_n $-lattice codes of full diversity

School of Sciences, Department of Mathematics, São Paulo State University - UNESP, Bauru, SP 17033-360, BR

^* Corresponding author
^* Corresponding author

Received: July 31, 2020

Revised: August 31, 2020

Early access: November 2020

Published: August 2022

This work was supported by FAPESP 2013/25977-7 and CNPq 429346/2018-2

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Abstract

Some important properties of lattices are packing density and full diversity, which may be good for signal transmission over both Gaussian and Rayleigh fading channel, respectively. The algebraic lattices are constructed through twisted homomorphism of some modules in the ring of integers of a number field $ \mathbb{K} $. In this paper, we present a construction of some families of rotated $ A_n- $lattices, for $ n = 2^{r-2}-1 $, $ r \geq 4 $, via totally real subfield of cyclotomic fields. Furthermore, closed-form expressions for the minimum product distance of those lattices are obtained through algebraic properties.

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Mathematics Subject Classification: Primary: 52C07; Secondary: 11H31, 11H71.

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Table 1. Normalized minimum product distance versus center density (from [5,12,15,16,18,19] and the results presented here)

$r $	$n $	$\sqrt[n]{d_{p}(\mathbb{Z}^n)} $	$\sqrt[n]{d_{p}(D_n)} $	$\sqrt[n]{d_{p}(A_n)} $	$\delta(\mathbb{Z}^n) $	$\delta(D_n) $	$\delta(A_n) $
$4 $	$3 $	$0.52275 $	$0.41491 $	$0.44544 $	$0.12500 $	$0.17677 $	$0.17677 $
$5 $	$7 $	$0.30080 $	$- $	$0.27602 $	$0.00780 $	$0.04419 $	$0.03125 $
$6 $	$15 $	$0.20138 $	$0.19229 $	$0.18513 $	$0.00003 $	$0.00276 $	$0.00138 $
$7 $	$31 $	$0.06220 $	$- $	$0.12782 $	$10^{-10} $	$10^{-5} $	$10^{-6} $
$8 $	$63 $	$0.09221 $	$0.09120 $	$0.08936 $	$10^{-19} $	$10^{-10} $	$10^{-11} $
$9 $	$127 $	$0.04542 $	$- $	$0.06284 $	$10^{-39} $	$10^{-20} $	$10^{-21} $
$10 $	$255 $	$0.03172 $	$- $	$0.04431 $	$10^{-77} $	$10^{-39} $	$10^{-40} $
$11 $	$511 $	$0.01819 $	$- $	$0.03129 $	$10^{-154} $	$10^{-78} $	$10^{-79} $
$12 $	$1023 $	$0.01569 $	$- $	$0.02211 $	$10^{-308} $	$10^{-155} $	$10^{-152} $
$13 $	$2047 $	$0.00522 $	$- $	$0.01563 $	$10^{-617} $	$10^{-309} $	$10^{-310} $
$14 $	$4095 $	$0.01106 $	$0.01106 $	$0.01106 $	$10^{-1233} $	$10^{-617} $	$10^{-619} $
$15 $	$8191 $	$0.00163 $	$- $	$0.00781 $	$10^{-2466} $	$10^{-1234} $	$10^{-1235} $
$16 $	$16383 $	$0.00319 $	$- $	$0.00552 $	$10^{-4932} $	$10^{-2467} $	$10^{-2468} $
$17 $	$32767 $	$0.00130 $	$- $	$0.00390 $	$10^{-9864} $	$10^{-4933} $	$10^{-4935} $
$18 $	$65535 $	$0.00276 $	$0.00276 $	$0.00276 $	$10^{-19729} $	$10^{-9865} $	$10^{-9867} $
$19 $	$131071 $	$0.00079 $	$- $	$0.00195 $	$10^{-39457} $	$10^{-19729} $	$10^{-19731} $
$20 $	$262143 $	$0.00138 $	$0.00138 $	$0.00138 $	$10^{-78913} $	$10^{-39457} $	$10^{-39460} $

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