New Results on Periodic Golay Pairs

2024-08-28

arXiv.org

AI Summary

Exhaustive enumeration of periodic Golay pairs (PGPs) has been limited to lengths $v leq 34$ due to combinatorial explosion. Method: We introduce a multi-level sequence compression/decompression strategy, integrate ordered generation principles to design an efficient isomorphism-free enumeration algorithm, and implement a high-performance parallel search framework. Contribution/Results: We complete the first full isomorphism-free enumeration for all $v leq 72$ (<10 CPU-years), and construct the first known PGP of length $v = 90$. Empirical analysis yields a new structural conjecture on PGP existence, verified for all $v < 100$. Consequently, the smallest open length for PGP existence is advanced from $v = 90$ to $v = 106$, substantially expanding the known existence boundary for this class of complementary sequences.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Models for Web evolution

Abstract

In this paper, we provide algorithmic methods for conducting exhaustive searches for periodic Golay pairs. Our methods enumerate several lengths beyond the currently known state-of-the-art available searches: we conducted exhaustive searches for periodic Golay pairs of all lengths $v leq 72$ using our methods, while only lengths $v leq 34$ had previously been exhaustively enumerated. Our methods are applicable to periodic complementary sequences in general. We utilize sequence compression, a method of sequence generation derived in 2013 by Djokovi'c and Kotsireas. We also introduce and implement a new method of"multi-level"compression, where sequences are uncompressed in several steps. This method allowed us to exhaustively search all lengths $v leq 72$ using less than 10 CPU years. For cases of complementary sequences where uncompression is not possible, we introduce some new methods of sequence generation inspired by the isomorph-free exhaustive generation algorithm of orderly generation. Finally, we pose a conjecture regarding the structure of periodic Golay pairs and prove it holds in many lengths, including all lengths $v lt 100$. We demonstrate the usefulness of our algorithms by providing the first ever examples of periodic Golay pairs of length $v = 90$. The smallest length for which the existence of periodic Golay pairs is undecided is now $106$.

Problem

• Develops algorithmic methods for exhaustive searches
• Extends periodic Golay pair enumeration to length 72
• Introduces multi-level compression to reduce computational requirements

Innovation

• Algorithmic exhaustive search using sequence compression
• Multi-level compression for efficient enumeration
• New sequence generation methods for complementary pairs