62 problems

Let the lower and upper bounds on the second-order secrecy rate be denoted by the expressions in Theorems 13 and 14, respectively. Second-order secrecy-rate bound conjecture. The l…

Consider the LLR-based decoder for ABS+ polar codes and the formulas from the lemma giving approximate LLRs. Redundant-operations conjecture. Naively applying these formulas in the…

Let the originally presented successive-cancellation (SC) decoding algorithm for ABS+ polar codes be given. Redundancy conjecture. Lots of calculations in this algorithm are redund…

QUP polar-code functional inequality. Then

The successive-cancellation tree decoder and the neural polar decoder are being compared for deletion-channel codes, and the neural decoder appears to attain slightly better frame-…

The trellis-based decoder and the neural polar decoder are compared through their frame error rates at various block lengths, with both decoders theoretically expected to perform s…

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…

Let $(n)$ be a sequence of polar codes with increasing blocklength $n$, fixed error-probability upper bound $P_e$, rate $\frac{r}{n}$, and channel capacity $C$. The scaling exponen…

The obtained kernels have a processing complexity for which column permutations may provide further improvements. Processing-complexity conjecture. The processing complexity of the…

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…

Let a code use an $m$-bit CRC and let its decoder have a maximum list size. CRC list-size conjecture. A maximum list size of around $2^m$ is necessary for the code to achieve perfo…

Optimality conjecture. The polar codes constructed by these two algorithms are optimal in terms of minimizing $cmathrm{FER}$ under the given successive-cancellation list decoding s…

SCAN absorption-set conjecture. The absorption set under SCAN decoding is limited to the trivial permutation.

Let $n=2^m$, let $\mathbf{G}_{n}^{\operatorname{RM}}$ be the Reed–Muller generator matrix, and consider the successive decoder for the associated code construction. The message bit…

Let polar codes and Reed–Muller (RM) codes be code families whose generator matrices are formed from rows of the same square matrix. Both families achieve capacity on binary memory…

Optimal-coordinate conjecture. This is the optimal way to find the coordinate that contributes the most to the error coefficient $A_{d_{min}}$. The conjecture is motivated by the p…

Consider a polar code with automorphisms used in automorphism-en ensemble (AE) decoding, and denote by $[\mathbbm{1}]$ the equivalence class of the identity automorphism. AE-decode…

Let the $(32,23)$ polar code be defined by $\mathcal{I}_{min}=\{7,9\}$. Suppose its affine automorphism group is $\mathsf{BLTA}(3,2)$, and let $[\mathbbm{1}]=\mathsf{BLTA}(3,1,1)$.…

BLTA characterization conjecture.

Affine automorphism conjecture. This inclusion should be an equality:

Let $W$ be a binary symmetric channel, and let $d707$ denote its scaling exponent, defined as the exponent governing the gap to capacity under polar-code construction:…

Quantization-only performance-loss conjecture. The performance degradation of the GA-based quantized decoder is due to the Gaussian approximation itself not providing a precise app…

Let $K_{\text{eNBCH}}$ denote the eNBCH kernels of sizes $16$ and $32$, and let the proposed algorithm produce permutations for these kernels. Optimality conjecture. Although the p…

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…

Rate-0 and Rate-1 pruning conjecture. Asymptotically, most of the savings in latency comes from pruning Rate-0 and Rate-1 nodes.