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In this paper, the discrete memoryless arbitrarily varying broadcast channel (AVBC) with receiver side information is studied and its random code and deterministic code capacity regions are derived for the average error criterion. In addition, it is analyzed for deterministic list codes and it is shown that the corresponding list capacity region displays a behavior, which is similar to Ahlswede’s famous dichotomy result for the single-user arbitrarily varying channel: it either equals the random code capacity region or otherwise has an empty interior. This is characterized in terms of list sizes at the receivers and an appropriate concept of symmetrizability for the AVBC with receiver side information. The scenario studied here is motivated by the broadcast phase of bidirectional relaying, where a half-duplex relay node establishes a bidirectional communication between two other nodes using a decode-and-forward protocol. The relay decodes the messages both nodes have sent in the initial multiple access phase and broadcasts a re-encoded composition of them in the succeeding broadcast phase. Then, the broadcast phase corresponds to the AVBC with receiver side information, which differs from the classical broadcast channel, since both receivers can exploit their own messages from the previous phase as side information for decoding. [ABSTRACT FROM PUBLISHER]

The sumset and inverse sumset theories of Freiman, Plünnecke, and Ruzsa, give bounds connecting the cardinality of the sumset \(A+B=\{a+b;\;a\in A,\,b\in B\}\) of two discrete sets \(A,B\) , to the cardinalities (or the finer structure) of the original sets \(A,B\) . For example, the sum-difference bound of Ruzsa states that, \(|A+B|\,|A|\,|B|\leq |A-B|^3\) , where the difference set \(A-B= \{a-b;\;a\in A,\,b\in B\}\) . Interpreting the differential entropy \(h(X)\) of a continuous random variable \(X\) as (the logarithm of) the size of the effective support of \(X\) , the main contribution of this paper is a series of natural information-theoretic analogs for these results. For example, the Ruzsa sum–difference bound becomes the new inequality, \(h(X+Y)+h(X)+h(Y)\leq 3h(X-Y)\) , for any pair of independent continuous random variables \(X\) and \(Y\) . Our results include differential-entropy versions of Ruzsa’s triangle inequality, the Plünnecke–Ruzsa inequality, and the Balog–Szemerédi–Gowers lemma. In addition, we give a differential entropy version of a Freiman–type inverse-sumset theorem, which can be seen as a quantitative converse to the entropy power inequality. Versions of most of these results for the discrete entropy \(H(X)\) were recently proved by Tao, relying heavily on a strong, functional form of the submodularity property of \(H(X)\) . Since differential entropy is not functionally submodular, in the continuous case many of the corresponding discrete proofs fail, in many cases requiring substantially new proof strategies. We find that the basic property that naturally replaces the discrete functional submodularity, is the data processing property of mutual information. [ABSTRACT FROM PUBLISHER]

In this paper, we study the capacity/entropy region of finite, directed, acyclic, multiple-sources, and multiple-sinks network by means of group theory and entropy vectors coming from groups. There is a one-to-one correspondence between the entropy vector of a collection of \(n\) random variables and a certain group-characterizable vector obtained from a finite group and \(n\) of its subgroups. We are looking at nilpotent group characterizable entropy vectors and show that they are all also abelian group characterizable, and hence they satisfy the Ingleton inequality. It is known that not all entropic vectors can be obtained from abelian groups, so our result implies that to get more exotic entropic vectors, one has to go at least to soluble groups or larger nilpotency classes. The result also implies that Ingleton inequality is satisfied by nilpotent groups of bounded class, depending on the order of the group. [ABSTRACT FROM AUTHOR]

Shannon's entropy power inequality can be viewed as characterizing the minimum differential entropy achievable by the sum of two independent random variables with fixed differential entropies. The entropy power inequality has played a key role in resolving a number of problems in information theory. It is therefore interesting to examine the existence of a similar inequality for discrete random variables. In this paper, we obtain an entropy power inequality for random variables taking values in a group of order 2^n , i.e., for such a group G , we explicitly characterize the function f_{G}(x,y) giving the minimum entropy of the sum of two independent G$ -valued random variables with respective entropies x$ and y$ . Random variables achieving the extremum in this inequality are thus the analogs of Gaussians in this case, and these are also determined. It turns out that is convex in $x$ for fixed $y$ and, by symmetry, convex in $y$ for fixed $x$ . This is a generalization to groups of order 2^n of the result known as Mrs. Gerber's Lemma. [ABSTRACT FROM AUTHOR]

The entropy power inequality (EPI) yields lower bounds on the differential entropy of the sum of two independent real-valued random variables in terms of the individual entropies. Versions of the EPI for discrete random variables have been obtained for special families of distributions with the differential entropy replaced by the discrete entropy, but no universal inequality is known (beyond trivial ones). More recently, the sumset theory for the entropy function yields a sharp inequality H(X+X^\prime)-H(X)\geq1\over2-o(1) when X , X^{\prime} are independent identically distributed (i.i.d.) with high entropy. This paper provides the inequality H(X+X^{\prime})-H(X)\geq g(H(X)) , where X , X^{\prime} are arbitrary i.i.d. integer-valued random variables and where g is a universal strictly positive function on \BBR+ satisfying g(0)=0 . Extensions to nonidentically distributed random variables and to conditional entropies are also obtained. [ABSTRACT FROM AUTHOR]

In their recent paper “Canalizing Boolean Functions Maximize Mutual Information,” Klotz et al. argued that canalizing Boolean functions maximize certain mutual informations by an argument involving Fourier analysis on the hypercube. This note supplies short new proofs of their results based on a coupling argument and also clarifies a point on the necessity of considering randomized functions. [ABSTRACT FROM AUTHOR]

A class of cognitive interference channels with state are investigated, in which a primary transmitter sends a message to two receivers (receivers 1 and 2) with assistance of a cognitive transmitter (that knows the primary transmitter’s message), and the cognitive transmitter also sends a separate message to receiver 2. The channel is corrupted by an independent and identically distributed state sequence. The scenario, in which the state sequence is noncausally known at both the cognitive transmitter and receiver 2, is first studied. The capacity region is obtained for both the discrete memoryless and Gaussian channels. The second scenario, in which the state sequence is noncausally known only at the cognitive transmitter, is further studied. Inner and outer bounds on the capacity region are obtained for the discrete memoryless channel and its degraded version. The capacity region is characterized for the degraded semideterministic channel and for channels that satisfy a less noisy condition. The Gaussian channels are further studied, which are partitioned into two cases based on how the interference compares with the signal at receiver 1. For each case, inner and outer bounds on the capacity region are derived, and partial boundaries of the capacity region are characterized. The full capacity region is also characterized for channels that satisfy certain conditions. It is shown that certain Gaussian channels achieve the capacity of the same channels with state noncausally known at both the cognitive transmitter and receiver 2. [ABSTRACT FROM PUBLISHER]