Abstract
A lattice in \(\mathbb {R}^{n}\) is a set of points (vectors) composed by all integer linear combinations of independent vectors.
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Notes
- 1.
Some authors use the row convention of considering basis vectors as rows of a generator matrix; we follow here the column convention.
- 2.
In several textbooks and papers, the minimum norm is defined as the square of this number.
- 3.
Strictly speaking, there might be more than one closest vector to y, which might cause ambiguities. In order for Q Λ (y) to be well defined, one has to break the ties, i.e., to decide which “faces” of the Voronoi cell to use. In order to simplify notation, we will avoid such a technicality and consider that ties are broken according to some well-defined systematic rule. Considering this rule we will also, by abuse of notation, sometimes say “the” closest lattice point to y. Notice that the faces of the Voronoi cell (i.e., the ambiguous points) have measure zero in \(\mathbb {R}^n\).
- 4.
Or approximately 1.53, in decibels. This number is sometimes referred to as the ultimate shaping gain of a lattice.
References
J.F. Adams, Lectures on Lie Groups (Midway Reprints Series) (University of Chicago Press, Chicago, 1983)
E. Agrell, T. Eriksson, A. Vardy, K. Zeger, Closest point search in lattices. IEEE Trans. Inf. Theory 48(8), 2201–2214 (2002)
M. Ajtai, Generating hard instances of lattice problems, in Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, STOC ’96, New York, NY (ACM, 1996), pp. 99–108
W. Banaszczyk, New bounds in some transference theorems in the geometry of numbers. Math. Ann. 296(1), 625–635 (1993)
E.S. Barnes, G.E. Wall, Some extreme forms defined in terms of abelian groups. J. Aust. Math. Soc. 1(1), 47–63 (1959)
N. Bourbaki, Lie Groups and Lie Algebras: Chapters 4–6 (Elements of Mathematics) (Springer, Berlin, 2002)
J.W.S. Cassels, An Introduction to the Geometry of Numbers (Springer, Berlin, 1997)
H. Cohen, A Course in Computational Algebraic Number Theory (Springer, New York, 1996)
J. Conway, N. Sloane, Voronoi regions of lattices, second moments of polytopes, and quantization. IEEE Trans. Inf. Theory 28(2), 211–226 (1982)
J.H. Conway, N.J.A. Sloane, Laminated lattices. Ann. Math. 116(3), 593–620 (1982)
J.H. Conway, N.J.A. Sloane, On the voronoi regions of certain lattices. SIAM J. Algebr. Discret. Methods 5(3), 294–305, (1984)
J.H. Conway, N.J.A. Sloane, Sphere-Packings, Lattices, and Groups (Springer, New York, 1998)
D. Micciancio, S. Goldwasser, Complexity of Lattice Problems. The Kluwer International Series in Engineering and Computer Science, vol. 671 (Kluwer Academic Publishers, Boston, MA, 2002). A cryptographic perspective
D. Micciancio, O. Regev, Lattice-Based Cryptography. Post-Quantum Cryptography (Springer, Berlin, 2009)
G. Nebe, E.M. Rains, N.J.A. Sloane, A simple construction for the barnes-wall lattices, in Codes, Graphs, and Systems (Springer, Berlin, 2002), pp. 333–342
C.A. Rogers, Packing and Covering (Cambridge University Press, Cambridge, 1964)
A. Schurmann, F. Vallentin, Computational approaches to lattice packing and covering problems. Discret. Comput. Geom. 35(1), 73–116 (2006)
N.J.A. Sloane, The sphere packing problem. Doc. Math. Extra Volume ICM, 387–396 (1998)
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Costa, S.I.R., Oggier, F., Campello, A., Belfiore, JC., Viterbo, E. (2017). Lattices and Applications. In: Lattices Applied to Coding for Reliable and Secure Communications. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-67882-5_2
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