16 problems

Let $P$ be a discrete memoryless channel, and denote by $\mathcal{E}^{(1)}_{2,P}$ and $\mathcal{E}^{(2)}_{2,P}$ the one-hop and two-hop error exponents, respectively, for transmitt…

High-noise protocol conjecture. A more sophisticated protocol can make the multiplicative constant $3/4$ arbitrarily close to one in the high-noise limit $p=q$, $p\to 1/2$.

Upper-bound strictness conjecture. The optimal learning rate cannot attain

Let $X$textendash$Y$ and $X$textendash$Z$ be binary symmetric channels with crossover probability $p$, and let $R_0^*$ denote the minimum rate of the error-free $Z$textendash$Y$ li…

Common-message splitting conjecture. Splitting the common (forwarding) message further into two parts is not necessary for the CRC.

Consider a symmetric $(n,k,n)$ half-duplex MIMO relay channel, meaning that the source and destination each have $n$ antennas and the relay has $k$ antennas. Let $d^*(r)$ denote it…

Let an $(m,k,n)$ half-duplex MIMO relay channel (HD-RC) have $m$, $k$, and $n$ antennas at its three nodes, and let its diversity–multiplexing tradeoff (DMT) be compared with that…

Constant-gap conjecture for amplify-and-forward. Under this condition,

Let $C^+$ denote the deterministic-relay-channel capacity upper bound and let $R_{CF}$ denote the compress-and-forward achievable rate for the Gaussian relay channel. The channel p…

Let soft-DF-BP operate with codes confined to sequences of “good” codes. Let the conditions of the source's Theorem formalize sufficient conditions for a rate to be achievable with…

Let ${\mathcal{C}}$ be a code drawn from a sequence of “good” codes, and let ${\bf Y}_2$ be the channel output at the relay. Let ${\bf Y}^{\textrm{BP}}_2$ be the output of belief p…

Consider soft-DF-BP and other relay strategies whose underlying code ${\mathcal{C}}$ is restricted to a sequence of “good” codes. Good-code limitation conjecture. In practice, soft…

In soft decode-and-forward belief propagation (soft-DF-BP), let ${\bf Y}^{\textrm{BP}}_2$ be the relay's BP estimate and let ${\bf \hat Y}^{\textrm{BP}}_2$ be its quantized version…

Let a broadcast relay channel with common relay have input alphabets represented by $X,X_1,X_2$ and output alphabets $Y_1,Z_1,Y_2,Z_2$. In the degraded case, the channel satisfies…

Modified compress-and-forward DM-tradeoff conjecture. The modified compress-and-forward strategy can achieve the optimal DM-tradeoff in general.

Consider a bi-directional relay channel in which the two terminals transmit with power $P$ over additive white Gaussian noise of variance $\sigma^2$. The exchange rate is the rate…