On the Non-existence of Perfect Sequences with the Array Orthogonality Property

Abstract

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.

Theorems & Definitions (13)

• Definition 1: Array Orthogonality Property
• Theorem 1: Mow mow1993
• proof
• Corollary 1
• proof
• Theorem 2: Bömer and Antweiler bomer1992
• proof
• Lemma 1
• proof
• Theorem 3