Three-phase Golay sequence and array triads

TL;DR

This paper explores 3-phase Golay sequence and array triads, providing exhaustive search results up to length 24, new construction methods, and insights into their existence patterns, extending the understanding beyond 2-phase Golay sequences.

Contribution

It introduces new construction methods for 3-phase Golay array triads and explains their existence patterns through projections, expanding the theoretical framework.

Findings

Existence pattern is richer than 2-phase Golay sequences.

No 3-phase Golay triad exists with length ≡ 4 mod 6.

Provided two construction methods and projection techniques.

Abstract

3-phase Golay sequence and array triads are a natural generalisation of 2-phase Golay sequence pairs, yet their study has until now been largely neglected. We present exhaustive search results for 3-phase Golay sequence triads for all lengths up to 24, showing that the existence pattern is much richer than that for 2-phase Golay sequence pairs. We give an elementary proof that there is no 3-phase Golay sequence triad whose length is congruent to 4 modulo 6. We give two construction methods for 3-phase Golay array triads, and show how to project 3-phase Golay array triads to lower-dimensional Golay triads. In this way we explain much of the existence pattern found by exhaustive search.