Speaker
Description
It is widely acknowledged that the search for Legendre pairs of higher orders presents a significant challenge. Therefore, rather than engaging in traditional, computationally intensive combinatorial searches, one might consider exploring Legendre pairs via structured matrices.
In this talk, we explore Legendre pairs through matrix structures and use compressed sequences to search for higher-order Legendre pairs. Our findings begin with an analysis of the Legendre pair matrix equation and its characteristics for order $\ell = p$, where $p$ is a prime number. We then extend our search to construct Legendre pairs of composite orders $\ell$. Ultimately, we implement a matrix compression method for Legendre pairs to facilitate the search for composite orders through structured matrices. This compressed search enables us to search high-order Legendre pairs through a low complexity FFT-like algorithm, rather than the computationally extensive combinatorial search.