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Entropy 2017, 19(9), 235; https://doi.org/10.3390/e19090235

Entropies of Weighted Sums in Cyclic Groups and an Application to Polar Codes

1
Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544-1000, USA
2
Department of Electrical Engineering, Princeton University, Princeton, NJ 08544-1000, USA
3
Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA
4
Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
*
Author to whom correspondence should be addressed.
Received: 14 February 2017 / Revised: 5 April 2017 / Accepted: 5 April 2017 / Published: 7 September 2017
(This article belongs to the Special Issue Entropy and Information Inequalities)
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Abstract

In this note, the following basic question is explored: in a cyclic group, how are the Shannon entropies of the sum and difference of i.i.d. random variables related to each other? For the integer group, we show that they can differ by any real number additively, but not too much multiplicatively; on the other hand, for Z / 3 Z , the entropy of the difference is always at least as large as that of the sum. These results are closely related to the study of more-sums-than-differences (i.e., MSTD) sets in additive combinatorics. We also investigate polar codes for q-ary input channels using non-canonical kernels to construct the generator matrix and present applications of our results to constructing polar codes with significantly improved error probability compared to the canonical construction. View Full-Text
Keywords: entropy; more-sums-than-differences set; polar code entropy; more-sums-than-differences set; polar code
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Abbe, E.; Li, J.; Madiman, M. Entropies of Weighted Sums in Cyclic Groups and an Application to Polar Codes. Entropy 2017, 19, 235.

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