## Entropy under additive Bernoulli and spherical noises

##### Author(s)

Ordentlich, Or; Polyanskiy, Yury
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Let Z[superscript n] be iid Bernoulli (δ) and U[superscript n] be uniform on the set of all binary vectors of weight δ[superscript n] (Hamming sphere). As is well known, the entropies of Z[superscript n] and U[superscript n] are within O(√n). However, if X[superscript n] is another binary random variable independent of Z[superscript n] and U[superscript n], we show that H(X[superscript n]+U[superscript n]) and H(X[superscript n]+Z[superscript n]) are within O(√n) and this estimate is tight. The bound is shown via coupling method. Tightness follows from the observation that the channels x[superscript n]⟼x[superscript n]+U[superscript n] and x[superscript n]⟼x[superscript n]+Z[superscript n] have similar capacities, but the former has zero dispersion. Finally, we show that despite the √n slack in general, the Mrs. Gerber Lemma for H(X[superscript n]+U[superscript n]) holds with only an O(log n) correction compared to its brethren for H(X[superscript n]+Z[superscript n]). ©2019 Paper presented at the 2018 IEEE International Symposium on Information Theory (ISIT 2018), June 17-22, 2018, Vail, Colo.

##### Date issued

2018-06##### Department

Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science##### Journal

2018 IEEE International Symposium on Information Theory (ISIT)

##### Publisher

Institute of Electrical and Electronics Engineers (IEEE)

##### Citation

Ordentlich, Or, and Yury Polyanskiy, "Entropy under additive Bernoulli and spherical noises." 2018 IEEE International Symposium on Information Theory (ISIT 2018) (Piscataway, N.J.: IEEE, 2018): p. 521-25 doi 10.1109/ISIT.2018.8437589 ©2018 Author(s)

Version: Author's final manuscript

##### ISBN

978-1-5386-4780-6