47 problems

Let a 2-user multiple-input-single-output broadcast channel have finite-precision channel state information at the transmitter, meaning that the channel estimation error does not i…

Let a MISO broadcast channel be a multiple-input single-output broadcast channel, and suppose that the channel-state information at the transmitter has finite precision. Lapidoth's…

Let $(X,Y,Z)$ be the input and outputs of the binary skew-symmetric broadcast channel (BSSC), and let $U$ and $V$ be random variables such that $(U,V)\to X\to(Y,Z)$ forms a Markov…

Marton's conjecture. This rate region characterizes the capacity region of every general two-user classical broadcast channel. Marton's region is the best known achievable region f…

Cover's capacity-region conjecture. The capacity region consists of the triples of nonnegative rates $(R,R_Y,R_Z)$ satisfying

Let $C_{PF}(\mathbf{N},P_T)$ denote the proportional fair capacity of a Gaussian broadcast channel with noise vector $\mathbf{N}$ and total power $P_T$. Let $\preceq$ denote the co…

Multiplexing-gain-one conjecture. The multiplexing gain of the sum capacity is equal to $1$.

Let a broadcast channel (BC) be a communication channel with one transmitter and multiple receivers, and let a BC be degraded when the receivers' outputs form a degraded Markov cha…

Let $p_{Y,Z|X}$ be a broadcast channel, and let $\Sigma_i^H$ denote the $i$th level of the linear entropy hierarchy. The capacity region is the set of achievable rate pairs…

Let $x-t-z$ be a Markov chain, with $y=x+n$, where $x$ and $n$ are independent, $n\sim N(0,1)$, and the input satisfies $\mathbb{E}[x^2]=P$. Suppose that the bottleneck output $z$…

Consider the capacity region for the degraded broadcast channel with general message sets, and the inner bound obtained by successive decoding of the merged messages, where each re…

Consider the decentralized MIMO broadcast channel with $(M,N_1,N_2)=(6,3,3)$, where $k$ denotes the number of informed transmit antennas and $\operatorname{DoF}$ denotes the sum de…

The setting is a multi-antenna broadcast channel with partial channel-state information at the transmitter (CSIT), where conventional approaches adapt techniques designed for perfe…

Consider the three-common-receiver instance of the K-user deterministic broadcast channel with two groupcast messages, and let the achievable region be the one given by Theorem 4.…

SLS optimality conjecture. For the $K>3$ user setting, these bounds should be sufficient to identify a regime in which SLS is optimal.

Finiteness conjecture. Only finitely many of these bounds should be non-redundant; determining the precise set of redundant bounds, or proving that only finitely many bounds exist,…

High-beta dominance conjecture. For either metric, as $\beta\to\infty$, the bound of Theorem 3 is at least as tight as the maximum of the bounds of Theorems 1 and 2:

Consider the two-user ergodic fading broadcast channel and the inner bound on its capacity region obtained by Shamai and Tuninetti. Shamai–Tuninetti conjecture. Their inner bound i…

MISO broadcast-channel sum-gain conjecture. The optimal sum beamforming gain is

Nonattainability conjecture. For both bandwidth compression and bandwidth expansion, the outer bound cannot be achieved by any source-channel code.

Consider a two-receiver memoryless broadcast channel with states and receiver message side information (RMSI), together with the corresponding channel without RMSI. Let a transmiss…

Consider a private-message AWGN broadcast channel with $Q>3$ receivers, ordered by noise levels as $N_1\leq N_2\leq\cdots\leq N_Q$. A group leader is the common subgraph obtained f…

Consider a two-user MISO broadcast channel in which the transmitter has perfect CSIT for one user and no CSIT for the other. Tandon et al.'s conjecture. With fixed CSIT, the optima…

Consider the $K$-user MISO broadcast channel with alternating channel-state information at the transmitter (CSIT). For the marginal probabilities shown in figure (7), namely … patt…

Feedback-frequency conjecture. Increasing the number of users should decrease the feedback frequency required to achieve the relevant DoF performance, as suggested by the outer-bou…