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The work proves uniqueness of the capacity-achieving input distribution by leveraging total positivity of the binomial kernel, and establishes bounds on the support size: an improved upper bound of order n^2 (improving prior order n results) and a lower bound of order √n, along with statements about the support locations and probability values. Capacity bounds are also derived, showing capacity scales on the order of (1/2)log(n).","Binomial Channel: On the Capacity-Achieving Distribution and Bounds on the Capacity  \nIan Zieder􀀃 , Antonino Favano†, Luca Barletta†, and Alex Dytso􀀃􀀃  \n􀀃 New Jersey Institute of Technology, Newark, NJ 07102, USA. Email: [ihz3@njit.edu](ihz3@njit.edu)[ ](ihz3@njit.edu)† Politecnico di Milano, Milano, 20133, Italy. Email: {antonino.favano, [luca.barletta](luca.barletta}@polimi.it)[}](luca.barletta}@polimi.it)[@polimi.it](luca.barletta}@polimi.it)[ ](luca.barletta}@polimi.it)􀀃􀀃 Qualcomm Flarion Technologies, Bridgewater, NJ 08807, USA. Email: [odytso2@gmail.com](odytso2@gmail.com)  \narXiv :2401 . 128 18v 1 [ cs .IT] 23 Jan 2024  \nAbstract—This work considers a binomial noise channel. The paper can be roughly divided into two parts. The 􀀂rst part is concerned with the properties of the capacity-achieving distribution. In particular, for the binomial channel, it is not known if the capacity-achieving distribution is unique since the output space is 􀀂nite (i.e., supported on integers 0, . . . , n) and the input space is in􀀂nite (i.e., supported on the interval [0, 1]), and there are multiple distributions that induce the same output distribution. This paper shows that the capacity-achieving distribution is unique by appealing to the total positivity property of the binomial kernel. In addition, we provide upper and lower bounds on the cardinality of the support of the capacity-achieving distribution. Speci􀀂cally, an upper bound of order n2 is shown, which improves on the previous upper bound of order n due to Witsenhausen. Moreover, a lower bound of order √n is shown. Finally, additional information about the locations and probability values of the support points is established.  \nThe second part of the paper focuses on deriving upper and lower bounds on capacity. In particular, 􀀂rm bounds are established for all n that show that the capacity scales as 12 log(n).  \nI. INTRODUCTION  \nWe consider a channel for which the relationship between the input X ∈ [0 , 1] and the output Y ∈ {0,..., n} is described by the binomial distribution:  \nPY |X (y|x) = 􀀒 ny􀀓 xy (1 − x)n−y . (1)  \nIn this work, we are interested in studying the capacity of this channel as a function of the number of trials n, that is  \nC (n) = max I (X;Y ) . (2)  \nPX : X∈[0 , 1]  \nIn addition to studying capacity, we are also interested in studying properties of an optimal capacity-achieving distribution distribution denoted by PX ⋆ .  \nA. Literature Review  \nThe binomial channel naturally arises in molecular communications and the interested reader is referred to [1]–[4] and references therein. The channel is also useful in the study of the deletion channel [5], [6] .  \nThe capacity of the binomial channel was 􀀂rst considered in [7] where the authors used minimax redundancy theorem in [8] to argue that asymptotically the capacity scales as ~~1~~2 log n. The exact capacity for the n = 1 case was computed in [1] where binary distribution with support on {0 , 1} was shown  \nto be capacity-achieving. To the best of our knowledge, there are no 􀀂rm bounds on the capacity of the binomial channel.  \nProperties of the capacity-achieving distribution have also been looked at. For example, the authors of [1] have designed an algorithm for computing capacity and a capacityachieving distribution by using a dual representation of the maximization problem. It is also known that, by using the Witsenhausen technique [9], there exists a capacity-achieving distribution with at most n+1 mass points. We note, however, that the Witsenhausen technique does not guarantee that the optimal input distribution is unique. In fact, for the binomial channel, uniqueness has not been shown; note that uniqueness is important not just for theoretical purposes, but also for algorithmic purposes. A conventional way to show that the capacity-achieving distribution is unique is by establishing that the mutual information is a strictly concave function of the input distribution. However, as will be shown by a","cbCaivC5ddEXTn6H","https://ap.wps.com/l/cbCaivC5ddEXTn6H","pdf",323560,5,1,15,"English","en",105,"# Introduction\n## Literature Review\n## Outline and Contributions\n## Notation\n# Main Results: Capacity-Achieving Distributions\n## Symmetry of Capacity-Achieving Distributions\n## Discreteness and Mutual Information Properties\n## Support Point Locations and Probabilities\n## Support Cardinality Bounds\n# Main Results: Capacity Bounds","[{\"question\":\"What channel model does the paper study?\",\"answer\":\"The paper studies a binomial noise channel where the conditional distribution is P_{Y|X}(y|x)=C(n,y)x^y(1-x)^{n-y} with input X∈[0,1] and output Y∈{0,...,n}.\"},{\"question\":\"How does the paper prove the capacity-achieving distribution is unique?\",\"answer\":\"It establishes uniqueness by using the total positivity property of the binomial kernel, instead of relying on strict concavity of mutual information.\"},{\"question\":\"What are the established bounds on the capacity as n grows?\",\"answer\":\"Upper and lower capacity bounds are derived for all n, showing that capacity scales on the order of (1/2)log(n).\"}]",1784187704,38,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":28},"binomial-channel-capacity-achieving-distribution-and-bounds-on-capacity","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/binomial-channel-capacity-achieving-distribution-and-bounds-on-capacity/83438/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-25","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What channel model does the paper study?","Question",{"text":76,"@type":77},"The paper studies a binomial noise channel where the conditional distribution is P_{Y|X}(y|x)=C(n,y)x^y(1-x)^{n-y} with input X∈[0,1] and output Y∈{0,...,n}.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How does the paper prove the capacity-achieving distribution is unique?",{"text":81,"@type":77},"It establishes uniqueness by using the total positivity property of the binomial kernel, instead of relying on strict concavity of mutual information.",{"name":83,"@type":74,"acceptedAnswer":84},"What are the established bounds on the capacity as n grows?",{"text":85,"@type":77},"Upper and lower capacity bounds are derived for all n, showing that capacity scales on the order of 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