Quaternary Legendre Pairs of Lengths 42, 46, 52, 58, 66, 72, and 80

A quaternary Legendre pair of length ℓ is a pair of sequences over the fourth roots of unity whose periodic autocorrelations sum to −2 at every nonzero shift; each such pair yields a quaternary Hadamard matrix of order 2ℓ+2 and a binary Hadamard matrix of order 4ℓ+4. This work presents explicit quaternary Legendre pairs at lengths 42, 46, 52, 58, 66, 72, and 80, resolving seven of the fourteen lengths at most 100 left open by the 2025 published lists. At lengths 64, 70, and 76 the reversible four-block queues recorded by the searches were exhausted and contain no pair; these are statements about the searched families, not nonexistence results.

The record contains two files.

1. quaternary_legendre_pairs.pdf is the paper.
2. quaternary-legendre-pairs-release-v1.0.0.tar.gz is the complete release: the paper source, the frozen witness packages, the three queue-exhaustion certificates, the search programs, a SHA-256 manifest of everything, and a dependency-free verifier. Running python3 verify_qlp_pairs.py from the extracted archive checks all twelve printed pairs in under a second using only the Python standard library and exact integer arithmetic. The paper is arranged so its principal results can be confirmed from the printed digit strings alone, without executing any of the accompanying software.

Software in the archive is MIT-licensed; the paper and result data are CC-BY-4.0 (see LICENSES.md inside).