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

TL;DR

This paper proves that sequences with the Array Orthogonality Property cannot surpass the $n^2$ length bound, reinforcing Mow's conjecture for a significant class of sequences and highlighting the role of algebraic structures.

Contribution

It establishes the $n^2$ bound for sequences generated by polynomial and rational index functions with the AOP, and contrasts this with recent non-commutative constructions.

Findings

Sequences from polynomial index functions cannot exceed the $n^2$ bound.

Attempted geometric expansions lead to destructive phase scattering.

The results depend on the algebraic nature of the sequence construction.

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.