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.

翻译：三十余年来，对完美周期自相关序列的探索一直受到Mow猜想的制约，该猜想断言在$n$相位字母表上不存在长度超过$n^2$的完美序列。尽管对所有可能序列类的证明仍是一个未解决的问题，但本文针对依赖阵列正交性质的一类重要构造建立了界限。我们证明，由纯二元多项式索引函数生成的序列由于代数周期性无法突破$n^2$的Frank-Heimiller界。此外，我们将此结果推广至取整有理索引函数，证明任何试图几何扩展阵列维度的尝试都会不可避免地导致破坏性的分数相位散射。要消除这种散射，就必须严格迫使相位空间坍缩，从而重新确立$n^2$的限制。最后，我们界定了这些定理的适用范围，指出其根本依赖于交换代数，并与近期在非交换单位四元数上构造无界完美序列的研究成果进行了对比。