PreprintCC BY 4.0arXiv
New Records for the Hadamard Maximal Determinant Problem
Abstract
The Hadamard maximal determinant problem seeks an n× n matrix X with entries in {± 1}, which maximizes the determinant | det X | for a given order n ∈ ℕ. We report matrices attaining new record determinants for orders n≡ 3pmod4 between $103$ and $119$, as well as n=51. We provide a proof of optimality within a specific family of circulant--block constructions for X. For these orders, we recast the Hadamard problem as a sequence reconstruction problem from a pair of periodic autocorrelation functions, subject to certain arithmetic conditions. We describe several learning--based strategies for reconstruction and provide a comparison with classical simulated annealing--style approaches as a benchmark.
Related
- The maximal determinant of order 15 over the third roots of unity
- Colombo's Determinant Problem
- A Braak-Inspired Spectral Determinant Approach to a Dirac Bound-State Problem
- On Mok's uniqueness problem of holomorphic isometries from complex unit balls of maximal dimension into classical domains
- On a problem of Nathanson related to minimal asymptotic bases and maximal asymptotic nonbases
BibTeX
@misc{butbaia2026records,
title = {{New Records for the Hadamard Maximal Determinant Problem}},
author = {Giorgi Butbaia and Justin Tan and Pragatheeswaran Vipulanandan and Xiaoyu Huang and Toby Saunders-A'Court and Lucas Fagan and Davide Passaro and Michele Tarquini and Sergei Gukov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.22518},
eprint = {2608.22518},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.22518},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - New Records for the Hadamard Maximal Determinant Problem
AU - Giorgi Butbaia
AU - Justin Tan
AU - Pragatheeswaran Vipulanandan
AU - Xiaoyu Huang
AU - Toby Saunders-A'Court
AU - Lucas Fagan
AU - Davide Passaro
AU - Michele Tarquini
AU - Sergei Gukov
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2608.22518
UR - https://arxiv.org/abs/2608.22518
AB - The Hadamard maximal determinant problem seeks an n× n matrix X with entries in {± 1}, which maximizes the determinant | det X | for a given order n ∈ ℕ. We report matrices attaining new record determinants for orders n≡ 3pmod4 between $103$ and $119$, as well as n=51. We provide a proof of optimality within a specific family of circulant--block constructions for X. For these orders, we recast the Hadamard problem as a sequence reconstruction problem from a pair of periodic autocorrelation functions, subject to certain arithmetic conditions. We describe several learning--based strategies for reconstruction and provide a comparison with classical simulated annealing--style approaches as a benchmark.
ER -