28 problems

Let $\mathcal{K}(r,V)$ be the minimum of $\mathbb{E}[\Phi(\Pi)]$ over probability distributions $P_\Pi$ on $\mathbb{R}$ with mean $r$ and variance $V$, where $\Phi$ is the standard…

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

Tightness conjecture. The upper bound on the linearithmic capacity of deterministic identification over additive white Gaussian noise channels should be tight, and the use of typic…

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'…

Robust joint source-channel coding conjecture. If any joint source-channel code exists that achieves the target distortion on average, then there should also exist one that meets t…

Optimality conjecture. The quantity $\xi^*(R,T)$ reflects the best fraction of $E_1(R,T,\theta)$, and correspondingly of $E_2(R,T,\theta)$ in the stated sense, that any decoder una…

Let $\mathcal{T}_{si} = \left( \mathcal{S},\mathbb{B},F,W \right)$ be a sub-injective trellis channel, and let the code length satisfy … Let $\mathcal{P}_{s}$ denote the last-pair…

For the Gaussian channel with signal-to-noise ratio $gamma$, let the channel input be an infinite standard normal vector and use the squared Euclidean distance between the transmit…

The minimum-distance decoding quantities $S_{j,n}$, $V_{1,n}$, and $V_{2,n}$ are defined from the distances between codewords and channel outputs, and the stopping time is determin…

Let $S(P)$ denote the channel skewness for the Gaussian channel, let $Q^{-1}(epsilon)$ be the inverse Gaussian tail function at error probability $epsilon$, and consider the achiev…

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…

Layer-crossing conjecture.

Twin-Reed–Muller conjecture. The twin-Reed–Muller code is the Reed–Muller code; equivalently, for $i,j\in[n]$,

Let $W:\mathbb{F}_q\to\mathcal{Y}$ be a $q$-ary channel, and let $\overline{W}$ denote the symmetrized channel obtained by adding an independent uniform random variable to the inpu…

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 $\theta=\frac{\delta-\gamma}{1-2\gamma}$, where $\gamma$ and $\delta$ are the channel parameters for the unfair noisy channel, and let $h$ denote the binary entropy function. C…

Let $\overline{\vartheta}(n;R)$ denote the minimum probability of error achievable by a rate-$R$ code of blocklength $n$, in the worst case over jammer strategies, and let…

Cardinality conjecture. It was conjectured that choosing $U$ to have cardinality $q$ readily gives $(R,R)\in{\mathcal E}_f$. The conjecture concerns whether the auxiliary-variable…

Let $M^*(\mathcal{W};n,\varepsilon_n)$ denote the maximal number of messages transmissible over the quantum channel $\mathcal{W}$ using $n$ channel uses with error probability…

Optimality conjecture. The established sphere-packing result, including its refined polynomial prefactor, is optimal for general symmetric classical-quantum channels.

Maximality conjecture. The criterion $\mathcal{A}_{QS}^{n}$ maximizes the represented probability among all stable criteria for the $n$-fold binary asymmetric channels for every…

A binary asymmetric channel (BAC) has binary input and output alphabets and transition probabilities … where $0<p<q<\frac{1}{2}$. A channel and a metric are matched when, for every…

The symbol-decision successive-cancellation (SC) algorithm decodes polar codes by making decisions on symbols, whereas the bit-decision SC algorithm makes decisions one bit at a ti…

Let $W$ be a symmetric binary-input, discrete memoryless channel, and let $4\mu4$ denote the scaling exponent of polar codes, governing the gap to capacity at fixed error probabili…

Let $W$ be an asymmetric or singular discrete memoryless channel with input alphabet ${\cal X}$ and output alphabet ${\cal Y}$. For rate $R$, let $\bar{P}_{\mathrm{e},c}(n,R)$ deno…