6 problems

Super-exponential codebook-size conjecture. The codebook size for any continuous-alphabet channel should be a super-exponential function of $n$, namely

Consider a frequency-based channel with $n$ object types, input count vector $X^n=(X_1,\ldots,X_n)$ having total count $ng_n$, and output $Z^n=(Z_1,\ldots,Z_n)$ such that, conditio…

Mutual-information concavity conjecture. The mutual information $I(X_1,X_2;Y_1,Y_2,Y_3)$ is concave in $(T_1,T_2,T_3)$. The conjecture concerns the sensing-time allocation in the t…

Let a discrete-time Poisson channel (DTPC) operate under an average power constraint, and let a capacity-achieving input distribution be a distribution attaining the channel capaci…

Consider the Poisson channel … with power constraint $\tfrac{1}{n}\sum_{t=1}^n x[t] \leq n^\alpha$. This is the Poisson channel with increasing power (PCIP). A normalized rate…

Let $X$ be a non-negative random variable input to the scalar Poisson channel, and let $P$ and $Q$ be probability measures governing the input under the true and mismatched models,…