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Maryna Viazovska

Publications and source records attributed to Maryna Viazovska.

At least 19 recordsLinked to original sources
arXiv2026

Evolving Towards Better Codes: LLM-Guided Search for High-Distance Binary Linear Codes

Evolutionary program search driven by large language models (LLMs) has produced record-breaking constructions for open problems in combinatorics and beyond. We apply this approach to the longstanding problem of improving the best-known bounds for binary linear codes. Building on the EvoTune evolutionary framework and the ShinkaEvolve codebase, we introduce LinCodeEvolve, which evolves code-construction programs against an exact minimum-distance evaluator. A strategy loop combines diversity-driven search and expert supervision: when progress plateaus, new strategies are used to redirect the search. LinCodeEvolve discovers seven record-breaking codes, $[172,21,66]$, $[173,20,68]$, $[176,21,68]$, $[181,21,70]$, $[184,21,72]$, $[189,22,72]$ and $[200,21,77]$, six of which have concise quasi-cyclic descriptions. With standard code modification techniques, they improve $22$ entries of the tables. Every code is verified by exhaustive enumeration. These results suggest that LLM-guided search can help find improved codes and complement existing methods in coding theory.

cs.IT↗
arXiv2026

Mean Value for Random Ideal Lattices

We investigate the average number of lattice points within a ball for the $n$th cyclotomic number field, where the lattice is chosen at random from the set of unit determinant ideal lattices of the field. We show that this average is nearly identical to the average number of lattice points in a ball among all unit determinant random lattices of the same dimension. To establish this result, we apply the Hecke integration formula and subconvexity bounds on Dedekind zeta functions of cyclotomic fields. The symmetries arising from the roots of unity in an ideal lattice allow us to prove the existence of ideal lattice packings of dimension $φ(n)$ and density $n\cdot 2^{-φ(n)}(1+o(1))$ as $n$ goes to infinity.

math.NT↗
arXiv2026

Integral Matrices of Fixed Rank over Number Fields

We prove an asymptotic formula for the number of fixed rank matrices with integer coefficients over a number field K/Q and bounded norm. As an application, we derive an approximate Rogers integral formula for discrete sets of module lattices obtained from lifts of algebraic codes. This in turn implies that the moment estimates of random lattices with a number field structure also carry through for large enough discrete sets of module lattices.

math.NT↗
arXiv2026

The sphere packing problem in dimension 24

Building on Viazovska's recent solution of the sphere packing problem in eight dimensions, we prove that the Leech lattice is the densest packing of congruent spheres in twenty-four dimensions and that it is the unique optimal periodic packing. In particular, we find an optimal auxiliary function for the linear programming bounds, which is an analogue of Viazovska's function for the eight-dimensional case.

math.NT↗
arXiv2026

The stealthiest particle configurations in 8 and 24 dimensions

We show that the $E_8$ and Leech lattices generate the stealthiest isometry-invariant, locally square-integrable point processes of intensity $1$ in $\mathbb{R}^8$ and $\mathbb{R}^{24}$, respectively, and that they are the unique stealthiest processes. We also apply the proof technique to sphere packing and energy minimization, and we construct a $20$-dimensional periodic configuration that is stealthier than any known $20$-dimensional lattice with the same particle density. This periodic configuration is a formal dual of Vardy's $20$-dimensional sphere packing.

math.MG↗
arXiv2026

Progress in Formalizing Sphere Packing in Dimension 8

In 2016, Viazovska famously solved the sphere packing problem in dimension $8$, using modular forms to construct a 'magic' function satisfying optimality conditions determined by Cohn and Elkies in 2003. In March 2024, Hariharan and Viazovska launched a project to formalize this solution and related mathematical facts in the Lean Theorem Prover. A significant milestone was achieved in February 2026: the result was formally verified, with the final stages of the verification done by Math, Inc.'s autoformalization model 'Gauss'. We discuss the techniques used to achieve this milestone, reflect on the unique collaboration between humans and Gauss, and discuss project objectives that remain.

math.MG↗
arXiv2026

Polynomial Freiman-Ruzsa, Reed-Muller codes and Shannon capacity

In 1948, Shannon used a probabilistic argument to show the existence of codes achieving a maximal rate defined by the channel capacity. In 1954, Muller and Reed introduced a simple deterministic code construction based on polynomial evaluations, which was conjectured and eventually proven to achieve capacity. Meanwhile, polarization theory emerged as an analytic framework to prove capacity results for a variation of RM codes - the polar codes. Polarization theory further gave a powerful framework for various other code constructions, but it remained unfulfilled for RM codes. In this paper, we settle the establishment of a polarization theory for RM codes, which implies in particular that RM codes have a vanishing local error below capacity. Our proof puts forward a striking connection with the recent proof of the Polynomial Freiman-Ruzsa conjecture [40] and an entropy extraction approach related to [2]. It further puts forward a small orbit localization lemma of potential broader applicability in combinatorial number theory. Finally, a new additive combinatorics conjecture is put forward, with potentially broader applications to coding theory.

cs.IT↗
arXiv2025

Effective module lattices and their shortest vectors

We prove tight probabilistic bounds for the shortest vectors in module lattices over number fields using the results of arXiv:2308.15275. Moreover, establishing asymptotic formulae for counts of fixed rank matrices with algebraic integer entries and bounded Euclidean length, we prove an approximate Rogers integral formula for discrete sets of module lattices obtained from lifts of algebraic codes. This in turn implies that the moment estimates of arXiv:2308.15275 as well as the aforementioned bounds on the shortest vector also carry through for large enough discrete sets of module lattices.

math.NT↗
arXiv2025

Module lattices and their shortest vectors

We study the shortest vector lengths in module lattices over arbitrary number fields, with an emphasis on cyclotomic fields. In particular, we sharpen the techniques of arXiv:2308.15275v2 to establish improved results for the variance of the number of lattice vectors of bounded Euclidean norm in a random module lattice. We then derive tight probabilistic bounds for the shortest vector lengths for several notions of random module lattice.

math.NT↗
arXiv2025

Algorithm Discovery With LLMs: Evolutionary Search Meets Reinforcement Learning

Discovering efficient algorithms for solving complex problems has been an outstanding challenge in mathematics and computer science, requiring substantial human expertise over the years. Recent advancements in evolutionary search with large language models (LLMs) have shown promise in accelerating the discovery of algorithms across various domains, particularly in mathematics and optimization. However, existing approaches treat the LLM as a static generator, missing the opportunity to update the model with the signal obtained from evolutionary exploration. In this work, we propose to augment LLM-based evolutionary search by continuously refining the search operator - the LLM - through reinforcement learning (RL) fine-tuning. Our method leverages evolutionary search as an exploration strategy to discover improved algorithms, while RL optimizes the LLM policy based on these discoveries. Our experiments on combinatorial optimization tasks demonstrate that integrating RL with evolutionary search accelerates the discovery of superior algorithms, showcasing the potential of RL-enhanced evolutionary strategies for algorithm design.

cs.AI↗
arXiv2024

Moments of the number of points in a bounded set for number field lattices

We examine the moments of the number of lattice points in a fixed ball of volume $V$ for lattices in Euclidean space which are modules over the ring of integers of a number field $K$. In particular, denoting by $ω_K$ the number of roots of unity in $K$, we show that for lattices of large enough dimension the moments of the number of $ω_K$-tuples of lattice points converge to those of a Poisson distribution of mean $V/ω_K$. This extends work of Rogers for $\mathbb{Z}$-lattices. What is more, we show that this convergence can also be achieved by increasing the degree of the number field $K$ as long as $K$ varies within a set of number fields with uniform lower bounds on the absolute Weil height of non-torsion elements.

math.NT↗
arXiv2023

Six-dimensional sphere packing and linear programming

We prove that the Cohn-Elkies linear programming bound for sphere packing is not sharp in dimension 6. The proof uses duality and optimization over a space of modular forms, generalizing a construction of Cohn-Triantafillou to the case of odd weight and non-trivial character.

math.MG↗
arXiv2023

Hyperbolic Fourier series

In this article we explain the essence of the interrelation described in [PNAS 118, 15 (2021)] on how to write explicit interpolation formula for solutions of the Klein-Gordon equation by using the recent Fourier pair interpolation formula of Viazovska and Radchenko from [Publ Math-Paris 129, 1 (2019)]. We construct explicitly the sequence in $L^1 (\mathbb{R} )$ which is biorthogonal to the system $1$, $\exp ( i πn x)$, $\exp ( i πn/ x)$, $n \in \mathbb{Z} \setminus \{0\}$, and show that it is complete in $L^1 (\mathbb{R})$. We associate with each $f \in L^1 (\mathbb{R}, (1+x^2)^{-1} d x)$ its hyperbolic Fourier series $h_{0}(f) + \sum_{n \in \mathbb{Z}\setminus \{0\}}(h_{n}(f) e^{ i πn x} + m_{n}(f) e^{-i πn / x} )$ and prove that it converges to $f$ in the space of tempered distributions on the real line. Applied to the above mentioned biorthogonal system, the integral transform given by $U_φ (x, y):= \int_{\mathbb{R}} φ(t) \exp \left( i x t + i y / t \right) d t $, for $φ\in L^{1} (\mathbb{R})$ and $(x, y) \in \mathbb{R}^{2}$, supplies interpolating functions for the Klein-Gordon equation.

math.AP↗
arXiv2022

Universal optimality of the $E_8$ and Leech lattices and interpolation formulas

We prove that the $E_8$ root lattice and the Leech lattice are universally optimal among point configurations in Euclidean spaces of dimensions $8$ and $24$, respectively. In other words, they minimize energy for every potential function that is a completely monotonic function of squared distance (for example, inverse power laws or Gaussians), which is a strong form of robustness not previously known for any configuration in more than one dimension. This theorem implies their recently shown optimality as sphere packings, and broadly generalizes it to allow for long-range interactions. The proof uses sharp linear programming bounds for energy. To construct the optimal auxiliary functions used to attain these bounds, we prove a new interpolation theorem, which is of independent interest. It reconstructs a radial Schwartz function $f$ from the values and radial derivatives of $f$ and its Fourier transform $\widehat{f}$ at the radii $\sqrt{2n}$ for integers $n\ge1$ in $\mathbb{R}^8$ and $n \ge 2$ in $\mathbb{R}^{24}$. To prove this theorem, we construct an interpolation basis using integral transforms of quasimodular forms, generalizing Viazovska's work on sphere packing and placing it in the context of a more conceptual theory.

math.MG↗
arXiv2020

Fourier uniqueness in $\mathbb{R}^4$

We show an interrelation between the uniqueness aspect of the recent Fourier interpolation formula of Radchenko and Viazovska and the Heisenberg uniqueness study for the Klein-Gordon equation and the lattice-cross of critical density, studied by Hedenmalm and Montes-Rodriguez. This has been known since 2017.

math.FA↗
arXiv2020

Fourier interpolation on the real line

We use weakly holomorphic modular forms for the Hecke theta group to construct an explicit interpolation formula for Schwartz functions on the real line. The formula expresses the value of a function at any given point in terms of the values of the function and its Fourier transform on the set $\{0, \pm\sqrt{1}, \pm\sqrt{2}, \pm\sqrt{3},\dots\}$.

math.NT↗
arXiv2013

Well separated spherical designs

For each $N\ge C_dt^d$ we prove the existence of a well separated spherical $t$-design in the sphere $S^d$ consisting of $N$ points, where $C_d$ is a constant depending only on $d$.

math.MG↗