Open AccessConstruction of Complex Lattice Codes via Cyclotomic Fields
Author(s) -
Edmir Daniel Carvalho,
Antonio Aparecido de Andrade,
Tariq Shah,
Cibele Cristina Trinca
Publication year - 2022
Publication title -
trends in computational and applied mathematics
Language(s) - English
Resource type - Journals
ISSN - 2676-0029
DOI - 10.5540/tcam.2022.023.01.00033
Subject(s) - ring of integers , lattice (music) , polynomial ring , cyclotomic field , mathematics , combinatorics , algebraic number , ring (chemistry) , algebraic number field , unit (ring theory) , finite field , polynomial , root of unity , discrete mathematics , physics , quantum mechanics , mathematical analysis , chemistry , mathematics education , organic chemistry , acoustics , quantum
Through algebraic number theory and Construction $A$ we extend an algebraic procedure which generates complex lattice codes from the polynomial ring \mathbb{F}_{2}[x]/(x^{n}-1), where \mathbb{F}_{2}=\{0,1\}, by using ideals from the generalized polynomial ring \frac{\mathbb{F}_{2}[x,\frac{1}{2}\mathbb{Z}_{0}]}{((x^{\frac{1}{2}})^{n}-1)} through the ring of integers $\mathcal{O}_{\mathbb{L}}$ of the cyclotomic field \mathbb{L}=\mathbb{Q}(\zeta_{2^{s}}), where \zeta_{2^{s}} is a 2^{s}-th root of the unit, with s2.