New Records for the Hadamard Maximal Determinant Problem

The Hadamard Maximal Determinant problem seeks an $$\begin{gathered} n \\ timesn \end{gathered}$$ matrix $X$ with entries in $\pm 1$, which maximizes the determinant $$\begin{gathered} vert \\ detX \\ vert \end{gathered}$$ for a given order $$\begin{gathered} n \\ in \\ mathbbN \end{gathered}$$. We report matrices attaining new record determinants for orders $$\begin{gathered} n \\ equiv3 \\ pmod4 \end{gathered}$$ 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.