Rotated $D_n$-lattices in dimensions power of 3

Year 2021, Volume: 8 Issue: 3, 151 - 160, 15.09.2021

Agnaldo J. Ferrari Grasiele C. Jorge Antonio A. De Andrade

Abstract

In this work, we present constructions of families of rotated $D_n$-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 $\mathbb{Z} \oplus \mathcal{A}_{2}^{k}$ lattices, where $\mathcal{A}_{2}^{k}$ is a direct sum of $k=\frac{3^{r-1}-1}{2}$ copies of the $A_2$-lattice, using free $\mathbb{Z}$-modules in $\mathbb{Z}[\zeta_{3^{r}}+\zeta_{3^{r}}^{-1}]$.

Keywords

Lattices, Cyclotomic fields, Signal transmission

References

• [1] A. A. Andrade, C. Alves, T. B. Carlos, Rotated lattices via th cyclotomic field Q(2r ), International Journal of Applied Mathematics 19(3) (2006) 321-331.
• [2] E. Bayer-Fluckiger, Lattices and number fields, Contemporary Mathematics 241 (1999) 69-84.
• [3] E. Bayer-Fluckiger, Upper bounds for Euclidean minima of algebraic number fields, Journal of Number Theory 121(2) (2006) 305-323.
• [4] E. Bayer-Fluckiger, G. Nebe, On the Euclidean minimum of some real number fields, Journal de ThÃlorie des Nombres de Bordeaux 17(2) (2005) 437-454.
• [5] E. Bayer-Fluckiger, F. Oggier, E. Viterbo, New algebraic constructions of rotated Zn-lattice constellations for the Rayleigh fading channel, IEEE Transactions on Information Theory 50(4) (2004) 702-714.
• [6] E. Bayer-Fluckiger, I. Suarez, Ideal lattices over totally real number fields and Euclidean minima, Archiv der Mathematik 86(3) (2006) 217-225.
• [7] J. Boutros, E. Viterbo, C. Rastello, J. C. Belfiori, Good lattice constellations for both Rayleigh fading and Gaussian channels, IEEE Trans. Inform. Theory 42(2) (1996) 502-517.
• [8] J. H. Conway, N. J. A. Sloane, Sphere packings, lattices and groups, Springer-Verlag (1988).
• [9] A. J. Ferrari, A. A. Andrade, Constructions of rotated lattice constellations in dimensions power of 3, Journal of Algebra and its Applications 17(9) (2018) 1850175-1 to 17.
• [10] A. J. Ferrari, A. A. Andrade, R. R. Araujo, J. C. Interlando, Trace forms of certain subfields of cyclotomic fields and applications, Journal of Algebra Combinatorics Discrete Structures and Applications 7(2) (2020) 141-160.
• [11] J. C. Interlando, J. O. D. Lopes, T. P. N. Neto, The discriminant of Abelian number fields. Journal of Algebra and Its Applications 5 (2006) 35-41.
• [12] G. C. Jorge, A. J. Ferrari, S. I .R. Costa, Rotated Dn-lattices, Journal of Number Theory 132 (2012) 2397-2406.
• [13] G. C. Jorge, S. I. R. Costa, On rotated Dn-lattices constructed via totally real number fields, Archiv der Mathematik 100 (2013) 323-332.
• [14] P. Samuel, Algebraic theory of numbers, Hermann, Paris (1970).
• [15] I. Soprunov, Lattice polytopes in coding theory, Journal of Algebra Combinatorics Discrete Structures and Applications 2 (2) (2015) 85-94.
• [16] I. N. Stewart, D. O. Tall, Algebraic number theory, Chapman & Hall, London (1987).
• [17] L. C. Washington, Introduction to cyclotomic fields, Springer-Verlag, New York (1982).