On the Non-existence of Perfect Sequences with the Array Orthogonality Property
This addresses a long-standing conjecture in signal processing for researchers, but it is incremental as it focuses on specific classes and contrasts with non-commutative cases.
The paper tackles the problem of whether perfect sequences with the Array Orthogonality Property can exceed the n^2 length bound, showing that for sequences generated by bivariate polynomial or floored rational index functions, the bound holds due to algebraic periodicity and phase scattering, with no concrete numbers provided.
For over three decades, the pursuit of perfect periodic autocorrelation sequences has been constrained by Mow's conjecture, which posits that no perfect sequence over an $n$-phase alphabet can exist with a length greater than $n^2$. While a proof across all conceivable sequence classes remains an open problem, this paper establishes bounds for a prominent class of constructions relying on the Array Orthogonality Property (AOP). We show that sequences generated by pure bivariate polynomial index functions cannot exceed the $n^2$ Frank-Heimiller bound due to algebraic periodicity. Furthermore, we extend this result to floored rational index functions, proving that attempts to geometrically expand the array dimensions inherently result in destructive fractional phase scattering. Neutralising this scattering strictly forces a collapse of the phase space, re-establishing the $n^2$ limit. Finally, we define the boundaries of these theorems, noting their fundamental reliance on commutative algebras, and contrast them with recent sequence constructions demonstrating the existence of unbounded perfect sequences over non-commutative unit quaternions.