13 problems

Let $k$ and $\epsilon$ be fixed and let $n$ be odd. Let $\mathcal{A}_n$ denote the class of admissible, antisymmetric functions on $n$ bits, and let $\mathsf{MAJ}_m$ denote the maj…

Let $k$ and $\epsilon$ be fixed and let $n$ be odd. Write $\mathcal{A}_n$ for the class of admissible functions on $n$ bits, and let $\mathcal{P}_k(f;\epsilon)$ denote the success…

Consider LP decoding of a low-density-parity-check code, and let selected bits lie on a critical loop identified through the loop-calculus expansion. For each such bit, the decoder…

Consider linear-programming (LP) decoding as a large-signal-to-noise-ratio limit of belief-propagation decoding, and suppose that an LP modification is required to preserve the lin…

Let an instanton noise configuration cause a failure of the bare belief-propagation (BP) decoder, and consider the loop-calculus expansion of the partition function, whose zero-ord…

Consider a code ensemble with edge-perspective variable-node degree distribution satisfying $\lambda_2>0$, and let $l$ denote the number of density-evolution iterations. Although t…

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…

Large-degree variable-node mismatch conjecture. The decoder is in general able to overcome this mismatch for VNs with a large number of incoming messages.

Let a Levin–Wen (LW) model be the Hamiltonian model determined by a unitary fusion category, and consider its ground-state subspace. Levin–Wen error-correction conjecture. The grou…

Modified-rule error-floor conjecture. Although the two rules have similar performance in the waterfall region, the modified rule should have superior error-floor performance becaus…

Let $\mathcal{C}$ be an LDPC code, let $\mathcal{D}$ be a decoder diversity set, and let $\mathcal{E}^{[t]}(\Lambda^{(A,B)})$ be the union of the error sets associated with all tra…

Consider an iterative decoding algorithm whose density-evolution behavior is analyzed over several stages, and suppose the current sufficient conditions are those of Theorem … can…

Fossorier's conjecture. Owing to the inaccurate reliability information delivered at the last iteration of BP decoding, order-$p$ OSD in cascade connection with BP should result in…