Probability estimates for fading and wiretap channels from ideal class zeta functions

David  Karpuk; Anne-Maria  Ernvall-Hytönen; Camilla  Hollanti; Emanuele  Viterbo

doi:10.3934/amc.2015.9.391

Article Contents

2015, Volume 9, Issue 4: 391-413. Doi: 10.3934/amc.2015.9.391

This issue Previous Article Next Article An enumeration of the equivalence classes of self-dual matrix codes

Probability estimates for fading and wiretap channels from ideal class zeta functions

1.
Department of Mathematics and Systems Analysis, Aalto University, P.O. Box 11100, FI-00076
2.
Department of Mathematics and Statistics, University of Helsinki, FI-00014
3.
Department of Electrical and Computer Systems Engineering, Monash University, P.O. Box 35, Clayton, Victoria 3800

Received: March 2013

Published: November 2015

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Abstract

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.

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Mathematics Subject Classification: Primary: 11H71, 11M06; Secondary: 94B75, 94B70.

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