On the Scalability of Quasi-Complementary Sequence Sets: Quadratic and Cubic Laws

TL;DR

This paper establishes fundamental quadratic and cubic scaling laws for the size of quasi-complementary sequence sets, providing tight bounds and explicit constructions that demonstrate these laws are asymptotically optimal.

Contribution

The work introduces a geometric framework and explicit constructions that prove quadratic and cubic upper bounds on QCSS set size are tight, revealing a fundamental scalability barrier.

Findings

Asymptotically optimal QCSSs satisfy M ≤ (1+o(1))K^2N

Near-optimal QCSSs satisfy M ≤ (1+o(1))K^3N^2 for certain tightness factors

Explicit constructions achieve bounds, confirming the tightness of the scaling laws

Abstract

This work is concerned with the fundamental scaling laws of quasi-complementary sequence sets (QCSSs) by understanding how large the set size (denoted by $M$) can grow with the flock size ($K$) and the sequence length ($N$). We first establish a geometric framework that transforms a QCSS into a complex unit-norm codebook, through which and by exploiting the density thresholds of the codebooks, certain polynomial upper bounds of the QCSS set size are obtained. Sharp quadratic and cubic scaling laws are then introduced. Specifically, we show that asymptotically optimal QCSSs with tightness factor $\rho=1$ satisfy $M \le (1+o(1))K^2N$, while asymptotically near-optimal QCSSs satisfy $M \le (1+o(1))K^3N^2$ for $\rho < {(1+\sqrt{5})}/{2}$. To validate these upper bounds, we further propose explicit additive-character and mixed-character based constructions for QCSSs that achieve $M = K^2N + K$ and $M = K^3N^2 + 2K^2N + K$, respectively, thereby showing that the quadratic and cubic scaling laws are asymptotically tight. Our proposed constructions admit flexible parameter choices, and their maximum correlation estimates are shown to be tight through explicit extremal examples. Additionally, it is conjectured that the cubic scaling law is universal for all $1<\rho\le 2$, i.e., any asymptotically near-optimal QCSSs should satisfy $M \le (1+o(1))K^3N^2$. This identifies a fundamental cubic barrier for QCSS scalability.