Some results related to the conjecture by Belfiore and Sol\'e

In the first part of the paper, we consider the relation between kissing number and the secrecy gain. We show that on an $n=24m+8k$-dimensional even unimodular lattice, if the shortest vector length is $\geq 2m$, then as the number of vectors of length $2m$ decreases, the secrecy gain increases. We will also prove a similar result on general unimodular lattices. We will also consider the situations with shorter vectors. Furthermore, assuming the conjecture by Belfiore and Sol\'e, we will calculate the difference between inverses of secrecy gains as the number of vectors varies. We will show by an example that there exist two lattices in the same dimension with the same shortest vector length and the same kissing number, but different secrecy gains. Finally, we consider some cases of a question by Elkies by providing an answer for a special class of lattices assuming the conjecture of Belfiore and Sol\'e. We will also get a conditional improvement on some Gaulter's results concerning the conjecture.Comment: This paper contains the note http://arxiv.org/abs/1209.3573. However, there are several new results, including the results concerning a conjecture by Elkie

Nonasymptotic Probability Bounds for Fading Channels Exploiting Dedekind Zeta Functions

In this paper, new probability bounds are derived for algebraic lattice codes. This is done by using the Dedekind zeta functions of the algebraic number fields involved in the lattice constructions. In particular, it is shown how to upper bound the error performance of a finite constellation on a Rayleigh fading channel and the probability of an eavesdropper's correct decision in a wiretap channel. As a byproduct, an estimate of the number of elements with a certain algebraic norm within a finite hyper-cube is derived. While this type of estimates have been, to some extent, considered in algebraic number theory before, they are now brought into novel practice in the context of fading channel communications. Hence, the interest here is in small-dimensional lattices and finite constellations rather than in the asymptotic behavior

Probability Estimates for Fading and Wiretap Channels from Ideal Class Zeta Functions

In this paper, new probability estimates are derived for ideal lattice codes from totally real number fields using ideal class Dedekind zeta functions. In contrast to previous work on the subject, it is not assumed that the ideal in question is principal. In particular, it is shown that the corresponding inverse norm sum depends not only on the regulator and discriminant of the number field, but also on the values of the ideal class Dedekind zeta functions. Along the way, we derive an estimate of the number of elements in a given ideal with a certain algebraic norm within a finite hypercube. We provide several examples which measure the accuracy and predictive ability of our theorems.Comment: 24 pages. Extends our earlier arxiv submission arxiv.1303.347