{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:13:59Z","timestamp":1760242439864,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2017,7,15]],"date-time":"2017-07-15T00:00:00Z","timestamp":1500076800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"NUS Young Investigator Award","award":["R-263-000-B37-133"],"award-info":[{"award-number":["R-263-000-B37-133"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This paper investigates polar codes for the additive white Gaussian noise (AWGN) channel. The scaling exponent    \u03bc    of polar codes for a memoryless channel     q  Y | X      with capacity     I (  q  Y | X   )     characterizes the closest gap between the capacity and non-asymptotic achievable rates as follows: For a fixed     \u03b5 \u2208 ( 0 , 1 )    , the gap between the capacity     I (  q  Y | X   )     and the maximum non-asymptotic rate     R n *     achieved by a length-n polar code with average error probability    \u03b5    scales as     n  - 1 \/ \u03bc     , i.e.,     I  (  q  Y | X   )  -  R n *  = \u0398  (  n  - 1 \/ \u03bc   )     . It is well known that the scaling exponent    \u03bc    for any binary-input memoryless channel (BMC) with     I  (  q  Y | X   )  \u2208  ( 0 , 1 )      is bounded above by     4 . 714    . Our main result shows that     4 . 714     remains a valid upper bound on the scaling exponent for the AWGN channel. Our proof technique involves the following two ideas: (i) The capacity of the AWGN channel can be achieved within a gap of     O (  n  - 1 \/ \u03bc     log n   )     by using an input alphabet consisting of n constellations and restricting the input distribution to be uniform; (ii) The capacity of a multiple access channel (MAC) with an input alphabet consisting of n constellations can be achieved within a gap of     O (  n  - 1 \/ \u03bc   log n )     by using a superposition of     log n     binary-input polar codes. In addition, we investigate the performance of polar codes in the moderate deviations regime where both the gap to capacity and the error probability vanish as n grows. An explicit construction of polar codes is proposed to obey a certain tradeoff between the gap to capacity and the decay rate of the error probability for the AWGN channel.<\/jats:p>","DOI":"10.3390\/e19070364","type":"journal-article","created":{"date-parts":[[2017,7,18]],"date-time":"2017-07-18T03:45:16Z","timestamp":1500349516000},"page":"364","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Scaling Exponent and Moderate Deviations Asymptotics of Polar Codes for the AWGN Channel"],"prefix":"10.3390","volume":"19","author":[{"given":"Silas","family":"Fong","sequence":"first","affiliation":[{"name":"Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117583, Singapore"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5008-4527","authenticated-orcid":false,"given":"Vincent","family":"Tan","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117583, Singapore"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2017,7,15]]},"reference":[{"key":"ref_1","unstructured":"Cover, T.M., and Thomas, J.A. 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