The Taussky-Kruskemper method for construction of root lattices
DOI:
https://doi.org/10.13069/jacodesmath.v13i3.284Keywords:
Root lattices, Number field, Minimum product distanceAbstract
In this paper, we present a method to construct the root lattices based on the Taussky-Kruskemper method which computes the generator matrix of an integral lattice, given its Gram matrix. This yields an algebraic lattice, in the sense that the lattice is built via the embedding of a number field.
Received: 15 March 2023 | Accepted: 01 March 2026
Downloads
References
A. A. Andrade, C. Alves, and T. B. Carlos, Rotated lattices via the cyclotomic field $\mathbb{Q}(\zeta_{2^r})$, Int. J. Appl. Math. 19(3) (2006) 321–331.
A. A. Andrade, A. J. Ferrari, C. Alves, and T. B. Carlos, Lattices via cyclotomic fields in dimensions $2$ and $4$, Int. J. Appl. Math. 20(8) (2007) 1093–1103.
A. A. Andrade, A. J. Ferrari, C. W. O. Benedito, and S. I. R. Costa, Constructions of algebraic lattices, Comput. Appl. Math. 29(3) (2010) 1–13.
C. Batut, K. Belabas, D. Bernardi, H. Cohen, and M. Olivier, PARI/GP: A Software Package for Computer-Aided Number Theory.
E. Bayer-Fluckiger, Lattices and number fields, Contemp. Math. 241 (1999) 69–84.
E. Bayer-Fluckiger, Ideal lattices, in: G. Wüstholz, A Panorama of Number Theory or The View from Baker's Garden, Cambridge University Press, Cambridge (2002) 168–184.
E. Bayer-Fluckiger and G. Nebe, On the Euclidean minimum of some real number fields, J. Théor. Nombres Bordeaux 17(2) (2005) 437–454.
E. Bayer-Fluckiger, F. Oggier, and E. Viterbo, New algebraic constructions of rotated $\mathbb{Z}^n$-lattice constellations for the Rayleigh fading channel, IEEE Trans. Inform. Theory 50(4) (2004) 702–714.
J. Boutros, E. Viterbo, C. Rastello, and J. C. Belfiore, Good lattice constellations for both Rayleigh fading and Gaussian channels, IEEE Trans. Inform. Theory 42(2) (1996) 502–518.
A. Campello and J. Strapasson, On sequences of projections of the cubic lattice, Comput. Appl. Math. 32(1) (2013) 57–69.
P. E. Conner and R. Perlis, A Survey of Trace Forms of Algebraic Number Fields, World Scientific (1984).
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, vol. 290 in Grundlehren der mathematischen Wissenschaften, Springer, New York (1999).
A. J. Ferrari, A. A. Andrade, R. R. Araujo, and J. C. Interlando, Trace forms of certain subfields of cyclotomic fields and applications, J Algebra Comb Discret. Struct Appl 7(2) (2020) 141–160.
A. J. Ferrari, A. A. Andrade, J. C. Interlando, and C. Alves, Characterizations of totally real subfields of 2-power cyclotomic fields and applications to signal set design, J Algebra Comb Discret. Struct Appl 11(2) (2024) 73–81.
A. J. Ferrari, G. C. Jorge, and A. A. Andrade, Rotated $D_n$-lattices in dimensions power of 3, J Algebra Comb Discret. Struct Appl 8(3) (2021) 151–160.
M. Kruskemper, Algebraic construction of bilinear forms over $\mathbb{Z}$, Pub. Math. de Besancon Th (1996) 97–1997.
D. Micciancio and S. Goldwasser, Complexity of lattice problems: A cryptographic perspective, The Springer Int. Ser. Eng. Comput. Sci. 671 (2002).
F. Oggier, Algebraic Methods for Channel Coding, Ph.D. thesis, Ecole Polytechnique Federale de Lausanne (EPFL), Lausanne, Switzerland (2005).
F. Oggier and E. Bayer-Fluckiger, Best rotated cubic lattice constellations for the Rayleigh fading channel, in: IEEE International Symposium on Information Theory (ISIT), IEEE, Yokohama, Japan (2003) 37–.
O. Taussky, On a theorem of latimer and macduffee, Canad. J. Math. 1 (1949) 300–302.
O. Taussky, On matrix classes corresponding to an ideal and its inverse, Illinois J. Math. 1(1) (1957) 108–113.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2026 Agnaldo José Ferrari
This work is licensed under a Creative Commons Attribution 4.0 International License.
