November  2015, 9(4): 391-413. doi: 10.3934/amc.2015.9.391

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, Finland, Finland

2. 

Department of Mathematics and Statistics, University of Helsinki, FI-00014, Finland

3. 

Department of Electrical and Computer Systems Engineering, Monash University, P.O. Box 35, Clayton, Victoria 3800, Australia

Received  March 2013 Published  November 2015

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.
Citation: David Karpuk, Anne-Maria Ernvall-Hytönen, Camilla Hollanti, Emanuele Viterbo. Probability estimates for fading and wiretap channels from ideal class zeta functions. Advances in Mathematics of Communications, 2015, 9 (4) : 391-413. doi: 10.3934/amc.2015.9.391
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show all references

References:
[1]

, SAGE open source mathematics software system,, , ().   Google Scholar

[2]

IEEE International Conference on Communications, (2011), 1-5. doi: 10.1109/iccw.2011.5963544.  Google Scholar

[3]

IEEE Trans. on Comm., 61 (2013), 3396-3403. doi: 10.1109/TCOMM.2013.061913.120278.  Google Scholar

[4]

IEEE International Symposium on Information Theory and its Applications, (2010), 174-178. doi: 10.1109/ISITA.2010.5650095.  Google Scholar

[5]

IEEE Information Theory Workshop, (2010), 1-5. doi: 10.1109/CIG.2010.5592923.  Google Scholar

[6]

IEEE International Symposium on Information Theory, (2014), 966-970. doi: 10.1109/ISIT.2014.6874976.  Google Scholar

[7]

IEEE Information Theory Workshop, 2011. Google Scholar

[8]

IEEE International Congress on Ultra-Modern Telecommunications and Control Systems and Workshops, 2011. Google Scholar

[9]

C. Hollanti, E. Viterbo and D. Karpuk, Nonasymptotic probability bounds for fading channels exploiting Dedekind zeta functions,, preprint, ().   Google Scholar

[10]

Second edition. Graduate Texts in Mathematics, 110. Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0853-2.  Google Scholar

[11]

IEEE Trans. on Inf. Theory, 24 (1978), 451-456. doi: 10.1109/TIT.1978.1055917.  Google Scholar

[12]

Information Theory, IEEE Tran, pp (2015), p1, arXiv:1103.4086 (2013). doi: 10.1109/TIT.2015.2494594.  Google Scholar

[13]

Foundations and Trends in Communications and Information Theory, 1 (2004), 333-416. doi: 10.1561/0100000003.  Google Scholar

[14]

Designs, Codes, and Cryptography, 73 (2014), 425-440. doi: 10.1007/s10623-014-9935-7.  Google Scholar

[15]

IEEE International Symposium on Information Theory, (2011), 2831-2835. doi: 10.1109/ISIT.2011.6034091.  Google Scholar

[16]

IEEE Information Theory Workshop, (2011), 135-139. doi: 10.1109/ITW.2011.6089362.  Google Scholar

[17]

IEEE Trans. on Inf. Theory, 59 (2013), 6060-6082. doi: 10.1109/TIT.2013.2266396.  Google Scholar

[18]

IEEE International Symposium on Information Theory, (2012), 3038-3042. doi: 10.1109/ISIT.2012.6284119.  Google Scholar

[19]

Bell Syst. Tech. Journal, 54 (1975), 1355-1387. doi: 10.1002/j.1538-7305.1975.tb02040.x.  Google Scholar

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