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New Results on Periodic Golay Pairs

Tyler Lumsden, Ilias Kotsireas, Curtis Bright

TL;DR

This work addresses the challenge of exhaustively identifying periodic Golay pairs by developing a scalable pipeline based on sequence compression and orderly generation. It extends prior exhaustive enumerations to all lengths $v \le 72$ and demonstrates the feasibility of multi-level compression to manage large search spaces, including uncompression back to $v$-length sequences. The authors also introduce a $(v/2)$-compression conjecture governing the possible forms of compressions, proving it for all $v<100$ (with a potential exception at $v=50$) and validating the conjecture against extensive data. The practical impact includes new $90$-length examples and a clearer structural picture of periodic Golay pairs, enabling more comprehensive classification and guiding future constructions and proofs in combinatorial design and related areas.

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 Djoković 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 $106$.

New Results on Periodic Golay Pairs

TL;DR

This work addresses the challenge of exhaustively identifying periodic Golay pairs by developing a scalable pipeline based on sequence compression and orderly generation. It extends prior exhaustive enumerations to all lengths and demonstrates the feasibility of multi-level compression to manage large search spaces, including uncompression back to -length sequences. The authors also introduce a -compression conjecture governing the possible forms of compressions, proving it for all (with a potential exception at ) and validating the conjecture against extensive data. The practical impact includes new -length examples and a clearer structural picture of periodic Golay pairs, enabling more comprehensive classification and guiding future constructions and proofs in combinatorial design and related areas.

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 using our methods, while only lengths 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 Djoković 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 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 . We demonstrate the usefulness of our algorithms by providing the first ever examples of periodic Golay pairs of length . The smallest length for which the existence of periodic Golay pairs is undecided is now .
Paper Structure (19 sections, 9 theorems, 21 equations, 5 tables, 7 algorithms)

This paper contains 19 sections, 9 theorems, 21 equations, 5 tables, 7 algorithms.

Table of Contents

  1. Introduction
  2. Background on Periodic Golay Pairs
  3. Equivalence
  4. Compression
  5. Candidate Generation
  6. Integer Partitions
  7. Orderly Generation
  8. Hybrid Generation
  9. Matching
  10. Uncompression
  11. Multi-level Compression
  12. Description of Uncompression
  13. Equivalence Filtering
  14. Equivalence Class Generation
  15. List Filtering
  16. ...and 4 more sections

Key Result

Theorem 1

Suppose $(A, B)$ is a $\mathop{\mathrm{PG}}\nolimits(v)$. If $a = a_0 + \cdots + a_{v-1}$ and $b = b_0 + \cdots + b_{v-1}$, then

Theorems & Definitions (16)

  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Corollary 2
  • Lemma 1
  • Proof
  • Example 1
  • Theorem 3
  • Proof
  • Example 2
  • ...and 6 more