Maximum Determinants of Binary Hankel Matrices

Raoul Bianchetti; Payam Danesh

doi:10.59973/ipil.433

Authors

Raoul Bianchetti Information Physics Institute, Gosport, Hampshire, United Kingdom
Payam Danesh Department of Biosystems Engineering, University of Tehran, Iran

DOI:

https://doi.org/10.59973/ipil.433

Keywords:

Binary Hankel matrix, Persymmetric matrix, Maximal determinant, Exact enumeration, Desnanot--Jacobi identity

Abstract

An order- binary Hankel matrix is constant on antidiagonals and is therefore determined by a word of 2n-1 bits. This structure makes exhaustive optimization possible at moderate orders, although the determinant remains a nonlocal function and numerical searches alone cannot establish completeness. We determine the exact signed extrema at orders 16 and 17 by computer-assisted proofs covering all 2³¹ and 2³³ defining words. The computations combine fraction-free elimination, Desnanot–Jacobi condensation with exact fallbacks, and finite-field elimination with certified prime moduli and reversal-orbit reduction. At order 16 the extrema are ±245760, attained with multiplicities 16 and 14. At order 17 the maximum is 935533, attained twice, and the minimum is -641375, also attained twice. The respective nonsingularity counts are 2133222509 and 8556144305. Three complete computations agree at order 16 and two at order 17 on extrema, multiplicities, nonsingularity counts, moments, and word-sensitive fingerprints. The order-16 absolute extremizers admit a compact nearly periodic description, whereas each order-17 signed extremal class is a single reversal orbit. These results extend the currently recorded binary Hankel determinant data by two orders.

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