13 problems

Deletion Bound I tightness conjecture. If set $\mathcal{J}$ satisfies both cover condition and swap condition, then Algorithm (Deletion Bound I) can find the exact value of…

Incorrect-path splitting conjecture. Under the Gaussian approximation, for any incorrect path that survives at some unfrozen bit $u_i$, it will split at some stage within…

QUP polar-code functional inequality. Then

Let $L=2^m$ with $m\ge 3$, and let ${{\mathsf L}_{k-m}}$ denote the log-likelihood ratio at bit position $k-m$. For each ${\bf v}_{k-m+1}^k\in\{0,1\}^m$, let…

BLTA characterization conjecture.

Consider polar codes decoded by list decoding with a fixed list size $L$, and let $k$ denote the code dimension. Dimension-dependent SC-performance conjecture. The SC-performance o…

Consider two polar codes $A$ and $B$ with the same code length $n$ and message length $k$. Let ${\rm FER_{SC}}(\cdot)$ and ${\rm FER_{ML}}(\cdot)$ denote their frame-error rates un…

Consider two polar codes $A$ and $B$ with the same code length $n$ and message length $k$. Let ${\rm FER_{SC}}(\cdot)$ and ${\rm FER_{ML}}(\cdot)$ denote their frame-error rates un…

Asymptotic mixing-rate conjecture.

Concavity conjecture. The function $\overline{g}_{n}(x)$ is concave on $[0,1]$ for every $n$.

Correct-path survival conjecture. As $P_e(u_i)$ approaches zero, with high probability the current path survives at $u_i$ without splitting and $u_i$ is correctly decoded. If the s…

Necessity conjecture for preservation of $IS$. If $IS$ is preserved by the polarization process, then such a subspace $V_S$ must exist. The condition is presented as necessary for…

Possible polarization dichotomy. Fix a $0<\delta<1$. There exists a partition of $\mathcal{S}_n$ into two sets $\mathcal{S}'_n$ and…