11 problems

Simplex-code optimality conjecture. Then $\mathcal{C}$ solves Problem. The paper reports experimental evidence that simplex codes perform best among all codes with the same paramet…

Exponential-size conjecture. The number of such sequences is exponential in $n$.

Let $p_1^{\star,2}\approx0.15547$ and $p_2^{\star,2}\approx0.84453$ be the numerically determined optimal proportions of weight-$1$ and weight-$2$ columns, and let…

For each finite $d$, let $\bm{p}^{\star,d}$ denote the unique optimum of the convex optimization problem defining the minimum random access expectation. Closed-form nonexistence co…

Universal hyperbolic bound. For any partition $s_1+s_2=k$ with $\max\{s_1,s_2\}\geq 2$,

In the short-molecule regime, let $M$ be the number of molecules, $L$ the molecule length, $\beta$ the length parameter, and ${\cal X}$ the input alphabet. The log-cardinality of a…

For each strand index $j\in [N]$, noisy reads are aggregated by base-by-base majority voting over $\Sigma = \{\mathsf{A,C,G,T}\}$ to form a consensus sequence…

Consider matrices over a finite field whose columns have prescribed Hamming weights, and let $T_{\max}(G)$ denote the maximum expected number of sampled columns needed to span a st…

Let the rate-$1/2$ code family constructed by the authors of BSGY24 have block dimension $k$ and random-access expectation measured by the relevant quantity $T(k)$. BSGY rate-1/2 c…

General capacity conjecture. The capacity of an arbitrary multi-draw DNA storage channel is

Support conjecture. A probability mass function with support in $\mathcal{W}'$ may be capacity-achieving for the side-informed cDNA data-embedding channel; equivalently, the capaci…