Year: 2017
Type: article
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.
Source: Entropy
DOI: 10.3390/e19090235
Authors: Emmanuel Abbe, Jiange Li, Mokshay Madiman
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $3,137
Funder: National Science Foundation
Awards: CIF: Small:Analog-to-Analog Compression: Fundamental Limits and Constructive Schemes, CIF: Medium: Collaborative Research: Information Theory and Statistical Inference from Large-Alphabet Data, CAREER: An integrated probabilistic approach to discrete and continuous extremal problems via information theory, G6671297155
OpenAlex ID: W2296585621