Dual-Step Optimization for Binary Sequences with High Merit Factors

TL;DR

This work tackles the LABS problem for long binary sequences by introducing a dual-step algorithm that first uses GPU-accelerated self-avoiding walks with skew-symmetry and restriction classes to generate promising candidates, then applies a priority-queue-driven refinement to explore the full search space and further improve merit factors. The method yields new best-known binary sequences for lengths $450 \le L \le 527$ (except $518$), with statistically significant improvements over prior results and substantial average gains in $F$. Seeding attempts with Legendre constructions suggest constructive seeds can be limited by local minima, highlighting the strength of unrestricted second-phase search. Overall, the approach demonstrates a practical and scalable path to discovering long binary sequences with high merit factors for applications in communications, physics, chemistry, and cryptography.

Abstract

The problem of finding aperiodic low auto-correlation binary sequences (LABS) presents a significant computational challenge, particularly as the sequence length increases. Such sequences have important applications in communication engineering, physics, chemistry, and cryptography. This paper introduces a novel dual-step algorithm for long binary sequences with high merit factors. The first step employs a parallel algorithm utilizing skew-symmetry and restriction classes to generate sequence candidates with merit factors above a predefined threshold. The second step uses a priority queue algorithm to refine these candidates further, searching the entire search space unrestrictedly. By combining GPU-based parallel computing and dual-step optimization, our approach has successfully identified new best-known binary sequences for all lengths ranging from 450 to 527, with the exception of length 518, where the previous best-known value was matched with a different sequence. This hybrid method significantly outperforms traditional exhaustive and stochastic search methods, offering an efficient solution for finding long sequences with good merit factors.

Figures (6)

• Figure 1: Largest known merit factors ($F$) for $L \le 527$. Black symbols are optimal solutions from exhaustive searches Packebusch16, blue symbols are solutions from stochastic approaches Boskovic17Boskovic24dimitrov22. The highlighted range $450 \le L \le 527$ indicates where the focus of this work is.
• Figure 2: The arhitecture of SAW algorithm.
• Figure 3: Parallel self-avoiding walk inside a block.
• Figure 4: Workflow of the two-phase search, that combines the algorithms.
• Figure 5: Merit factor values ($F$) for binary sequences of length $450 \le L \le 527$. The dual-step approach found sequences with significantly higher merit factors compared to the previous best known values reported by Dimitrov dimitrov22. The merit factors of constructed sequences are also plotted for comparison.