12 problems

Close-packing uniqueness conjecture. This code is the only case for which a close-packing exists in dimension $n$.

Conjecture on coset-representative choices. There would be no big difference in performance between different choices of the set of coset representatives or of the mapping within t…

Gap-removal conjecture. This gap can be completely removed with tighter bounds for the variational distance between the discrete and continuous Gaussians.

Let $\delta>0$ denote the gap between the list-decoding capacity and the transmission rate. A lattice code is required to operate at rate $\frac12\log(P/N)-\delta$, and its list si…

Let $\delta\coloneqq\frac{1}{2}\log\frac{P}{N}-R$ measure the gap between the transmission rate $R$ and the list decoding capacity of the quadratically constrained adversarial chan…

For $n=2,3$ and $p=3,4$, consider linear quasi-perfect codes in $44^n$ under the $4ell_p$ metric, with codes identified up to congruence. The candidate codes are the lattices liste…

A mod-$\Lambda$ transformation applies receiver scaling followed by reduction modulo a lattice, producing a mod-$\Lambda$ channel; mutual information and error exponent are compare…

Let $L$ be a lattice in $M_4(\mathbb{C})\cap M_2(\mathbf{H})$, where the normalized minimum determinant is measured for the lattice code. Optimality conjecture. We conjecture that…

BB-code determinant decay conjecture. Asymptotically, for very large $N$, the determinants of the BB-code decay with exponent $\delta=2$. The paper gives the bounds…

Decoded-vector quadratic-form conjecture. The finally decoded $\underline{b}$, not only the $\underline{b}$ associated with the strongest consistent Gaussians, minimizes an express…

Quadratic-form conjecture. For $d>2$, the strongest consistent Gaussians likewise relate to an integer vector $\underline{b}$ that minimizes an expression of the form

A Gaussian message is consistent when it corresponds to a lattice-point Gaussian of the type analyzed in the decoder's convergence process; the decoder operates at noise conditions…