On the lower bound for kissing numbers of ℓp\ell_p-spheres in high dimensions

In this paper, we give some new lower bounds for the kissing number of $\ell_p$-spheres. These results improve the previous work due to Xu (2007). Our method is based on coding theory.Comment: 15 pages, 4 figures; any comments are welcom

Quantum Stabilizer Codes, Lattices, and CFTs

There is a rich connection between classical error-correcting codes, Euclidean lattices, and chiral conformal field theories. Here we show that quantum error-correcting codes, those of the stabilizer type, are related to Lorentzian lattices and non-chiral CFTs. More specifically, real self-dual stabilizer codes can be associated with even self-dual Lorentzian lattices, and thus define Narain CFTs. We dub the resulting theories code CFTs and study their properties. T-duality transformations of a code CFT, at the level of the underlying code, reduce to code equivalences. By means of such equivalences, any stabilizer code can be reduced to a graph code. We can therefore represent code CFTs by graphs. We study code CFTs with small central charge c = n ≤ 12, and find many interesting examples. Among them is a non-chiral E8 theory, which is based on the root lattice of E8 understood as an even self-dual Lorentzian lattice. By analyzing all graphs with n ≤ 8 nodes we find many pairs and triples of physically distinct isospectral theories. We also construct numerous modular invariant functions satisfying all the basic properties expected of the CFT partition function, yet which are not partition functions of any known CFTs. We consider the ensemble average over all code theories, calculate the corresponding partition function, and discuss its possible holographic interpretation. The paper is written in a self-contained manner, and includes an extensive pedagogical introduction and many explicit examples

Cavity approach to sphere packing in Hamming space

In this paper we study the hard sphere packing problem in the Hamming space by the cavity method. We show that both the replica symmetric and the replica symmetry breaking approximations give maximum rates of packing that are asymptotically the same as the lower bound of Gilbert and Varshamov. Consistently with known numerical results, the replica symmetric equations also suggest a crystalline solution, where for even diameters the spheres are more likely to be found in one of the subspaces (even or odd) of the Hamming space. These crystalline packings can be generated by a recursive algorithm which finds maximum packings in an ultra-metric space. Finally, we design a message passing algorithm based on the cavity equations to find dense packings of hard spheres. Known maximum packings are reproduced efficiently in non trivial ranges of dimensions and number of spheres.Comment: 23 pages, 11 figures; typos correcte

Multiple Packing: Lower Bounds via Error Exponents

We derive lower bounds on the maximal rates for multiple packings in high-dimensional Euclidean spaces. Multiple packing is a natural generalization of the sphere packing problem. For any $N>0$ and $L\in\mathbb{Z}_{\ge2}$, a multiple packing is a set $\mathcal{C}$ of points in $\mathbb{R}^n$ such that any point in $\mathbb{R}^n$ lies in the intersection of at most $L-1$ balls of radius $\sqrt{nN}$ around points in $\mathcal{C}$. We study this problem for both bounded point sets whose points have norm at most $\sqrt{nP}$ for some constant $P>0$ and unbounded point sets whose points are allowed to be anywhere in $\mathbb{R}^n$. Given a well-known connection with coding theory, multiple packings can be viewed as the Euclidean analog of list-decodable codes, which are well-studied for finite fields. We derive the best known lower bounds on the optimal multiple packing density. This is accomplished by establishing a curious inequality which relates the list-decoding error exponent for additive white Gaussian noise channels, a quantity of average-case nature, to the list-decoding radius, a quantity of worst-case nature. We also derive various bounds on the list-decoding error exponent in both bounded and unbounded settings which are of independent interest beyond multiple packing.Comment: The paper arXiv:2107.05161 has been split into three parts with new results added and significant revision. This paper is one of the three parts. The other two are arXiv:2211.04407 and arXiv:2211.0440