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Let $X^n$ be a uniformly distributed $n$-dimensional binary vector, and $Y^n$ be the result of passing $X^n$ through a binary symmetric channel (BSC) with crossover probability $\alpha$. A recent conjecture postulated by Courtade and Kumar…

We introduce a simply stated conjecture regarding the maximum mutual information a Boolean function can reveal about noisy inputs. Specifically, let $X^n$ be i.i.d. Bernoulli(1/2), and let $Y^n$ be the result of passing $X^n$ through a…

Suppose $\XX{N}$ is a uniformly distributed $N$-dimensional binary vector and $\YY{N}$ is obtained by passing $\XX{N}$ through a binary symmetric channel with crossover probability $\alpha$. Recently, Courtade and Kumar postulates that…

Suppose $X$ is a uniformly distributed $n$-dimensional binary vector and $Y$ is obtained by passing $X$ through a binary symmetric channel with crossover probability $\alpha$. A recent conjecture by Courtade and Kumar postulates that…

We prove the Courtade-Kumar conjecture, which states that the mutual information between any Boolean function of an $n$-dimensional vector of independent and identically distributed inputs to a memoryless binary symmetric channel and the…

Suppose $Y^{n}$ is obtained by observing a uniform Bernoulli random vector $X^{n}$ through a binary symmetric channel. Courtade and Kumar asked how large the mutual information between $Y^{n}$ and a Boolean function $\mathsf{b}(X^{n})$…

We prove the Courtade-Kumar conjecture, for several classes of n-dimensional Boolean functions, for all $n \geq 2$ and for all values of the error probability of the binary symmetric channel, $0 \leq p \leq 1/2$. This conjecture states that…

We prove the Courtade-Kumar conjecture, for certain classes of $n$-dimensional Boolean functions, $\forall n\geq 2$ and for all values of the error probability of the binary symmetric channel, $\forall 0 \leq p \leq \frac{1}{2}$. Let…

We consider a channel with a binary input X being corrupted by a continuous-valued noise that results in a continuous-valued output Y. An optimal binary quantizer is used to quantize the continuous-valued output Y to the final binary output…

A Boolean function $g$ is said to be an optimal predictor for another Boolean function $f$, if it minimizes the probability that $f(X^{n})\neq g(Y^{n})$ among all functions, where $X^{n}$ is uniform over the Hamming cube and $Y^{n}$ is…

The ability of information processing in biologically motivated Boolean networks is of interest in recent information theoretic research. One measure to quantify this ability is the well known mutual information. Using Fourier analysis we…

Let $(\mathbf X, \mathbf Y)$ denote $n$ independent, identically distributed copies of two arbitrarily correlated Rademacher random variables $(X, Y)$. We prove that the inequality $I(f(\mathbf X); g(\mathbf Y)) \le I(X; Y)$ holds for any…

We consider collocated wireless sensor networks, where each node has a Boolean measurement and the goal is to compute a given Boolean function of these measurements. We first consider the worst case setting and study optimal block…

We prove the "Most informative boolean function" conjecture of Courtade and Kumar for high noise $\epsilon \ge 1/2 - \delta$, for some absolute constant $\delta > 0$. Namely, if $X$ is uniformly distributed in $\{0,1\}^n$ and $Y$ is…

We address a sequential decision problem that arises in the computation of symmetric Boolean functions of distributed data. We consider a collocated network, where each node's transmissions can be heard by every other node. Each node has a…

We consider a channel with discrete binary input X that is corrupted by a given continuous noise to produce a continuous-valued output Y. A quantizer is then used to quantize the continuous-valued output Y to the final binary output Z. The…

A Boolean function of n bits is balanced if it takes the value 1 with probability 1/2. We exhibit a balanced Boolean function with a randomized evaluation procedure (with probability 0 of making a mistake) so that on uniformly random…

A Boolean function $f({\vec x})$ is sensitive to bit $x_i$ if there is at least one input vector $\vec x$ and one bit $x_i$ in $\vec x$, such that changing $x_i$ changes $f$. A function has sensitivity $s$ if among all input vectors, the…

We address the problem of finding optimal strategies for computing Boolean symmetric functions. We consider a collocated network, where each node's transmissions can be heard by every other node. Each node has a Boolean measurement and we…

We consider the function computation problem in a three node network with one encoder and two decoders. The encoder has access to two correlated sources $X$ and $Y$. The encoder encodes $X^n$ and $Y^n$ into a message which is given to two…