32 problems

Let $\mathcal{X}$ be an input alphabet, let $\mathcal{B}$ be an output quantum system, and let $x\mapsto\rho_{\mathcal{B}}^x$ be a classical–quantum channel. For an input distribut…

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…

Consider the channel coding reliability function, namely the best achievable exponential decay rate of the decoding error probability at each transmission rate, and Poor and Verdú'…

Joint source-channel coding (JSCC) combines source and channel coding for a discrete memoryless source and a discrete memoryless channel, whereas tandem coding performs source and…

Let a classical-quantum channel map each input symbol to an arbitrary, possibly mixed, quantum state, and consider random i.i.d. codes generated according to a fixed input distribu…

For a quantum channel simulation protocol, let the error exponent denote the best rate of exponential convergence of the simulation error toward perfect performance, as a function…

Consider a Gel'fand–Pinsker channel with a variable-rate random-binning code construction in which the binning rate depends on the empirical distribution of the side information, a…

Let $W\in{\mathcal CH}({\mathcal X},{\mathcal Y})$ be a channel and let $R$ be a rate such that $E(W,R)<+\infty$, where $E(W,R)$ is the channel reliability function and…

Let a discrete memoryless channel (DMC) be given, and consider a communication system using a generalized random Gilbert–Varshamov (RGV) code ensemble. Universality means that both…

Let a generalized random Gilbert–Varshamov (RGV) code ensemble be used for communication over a discrete memoryless channel, and let the typical random-coding (TRC) error exponent…

Positive-rate extension conjecture. The conclusion of Corollary 2 should extend from zero-rate code sequences to arbitrary code sequences of positive rate.

Let $W$ be an asymmetric or singular discrete memoryless channel, let $R$ be the coding rate, and let $\bar{P}_{\mathrm{e},\mathrm{c}}(N,R)$ denote the relevant constant-compositio…

Let $R$ denote the coding rate, let $R_{\text{c2}}$ be the second critical rate, and let $\tau$ be the parameter in the lower-bound optimization for the typical random code (TRC) e…

Deterministic-maximizer conjecture. For all $\alpha,\lambda\in\mathbb{R}_+$, the vectors $\mathbf{a}_\alpha^*$ and $\mathbf{b}_\alpha^*$ that maximize…

Finite-ratio single-channel conjecture. When $\alpha$ does not take extreme values, using one channel for the training sequence and another, possibly the same, channel for the sour…

Achievability-bound conjecture. $E_\text{r}(R)$ also yields an achievability bound on the error exponent for source coding with quantum side information.

Exponent-improvement conjecture. The achievability bound can be further improved to $E_\mathrm{r}$.

Exponential tightness conjecture. Except for these two possible improvements, the analysis of this specific scheme is exponentially tight.

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:

Rate-optimization interval conjecture. In the exact recursion, the upper endpoint $\min(1,2R)$ can be replaced by $R$; that is, the optimization may be restricted to

Concatenated polar coding error-exponent conjecture. The lower bound

Consider a variable-length feedback code with $M=2^K$ messages, target error probability $P_e$, expected decoding time $\mathbb{E}[T]$, rate $\overline{R}$, channel capacity $C$, f…

Let $W(z|x)$ be the single-letter transition probabilities of a memoryless channel, and consider the proposed universal decoder for channels whose transmitted codewords undergo los…

A mod-$\Lambda$ transformation applies receiver scaling followed by reduction modulo a lattice, producing a mod-$\Lambda$ channel; mutual information and error exponent are compare…

For a discrete memoryless channel, let the random-coding exponent be the exponent achieved by the standard random code ensemble, and let a random linear ensemble mean an ensemble o…