Quaternary Legendre Pairs of Lengths 42, 46, 52, 58, 66, 72, and 80
Abstract
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. Fourteen even lengths up to 100 remained open in the 2025 tables of Kotsireas, Koutschan and Winterhof and of Jedwab and Pender. We settle seven of them --- the four smallest open lengths, 42, 46, 52, and 58, together with 66, 72, and 80 --- and print every pair in full. Checking one takes a few lines of code and exact arithmetic over the Gaussian integers, so the main results can be confirmed without running any of our software. Each search writes a pair as four binary sequences under a Gray map, restricts to reflection-symmetric rows, imposes an exact spectral constraint at the midpoint frequency, and matches candidates meet-in-the-middle on their autocorrelation vectors; from length 52 onward, exact compression along divisor chains shrinks the search before any candidate is lifted. Everything is deterministic: no seed, heuristic stopping rule, or floating-point value enters an acceptance decision. At lengths 64, 70 and 76 the same machinery ran to exhaustion and found nothing: the reversible four-block queue built at each length contains no pair. These are exhaustions of a specific recorded queue, not nonexistence results, and the certificates do not by themselves establish that the queue captures every reversible object of its length. A closing section sorts every claim in the paper by what a reader must accept in order to believe it, and states exactly how far the accompanying artifacts support each one.
Related
BibTeX
@misc{lebedev2026quaternary,
title = {{Quaternary Legendre Pairs of Lengths 42, 46, 52, 58, 66, 72, and 80}},
author = {Nikita Lebedev},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04589},
eprint = {2609.04589},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04589},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Quaternary Legendre Pairs of Lengths 42, 46, 52, 58, 66, 72, and 80
AU - Nikita Lebedev
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.04589
UR - https://arxiv.org/abs/2609.04589
AB - 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. Fourteen even lengths up to 100 remained open in the 2025 tables of Kotsireas, Koutschan and Winterhof and of Jedwab and Pender. We settle seven of them --- the four smallest open lengths, 42, 46, 52, and 58, together with 66, 72, and 80 --- and print every pair in full. Checking one takes a few lines of code and exact arithmetic over the Gaussian integers, so the main results can be confirmed without running any of our software. Each search writes a pair as four binary sequences under a Gray map, restricts to reflection-symmetric rows, imposes an exact spectral constraint at the midpoint frequency, and matches candidates meet-in-the-middle on their autocorrelation vectors; from length 52 onward, exact compression along divisor chains shrinks the search before any candidate is lifted. Everything is deterministic: no seed, heuristic stopping rule, or floating-point value enters an acceptance decision. At lengths 64, 70 and 76 the same machinery ran to exhaustion and found nothing: the reversible four-block queue built at each length contains no pair. These are exhaustions of a specific recorded queue, not nonexistence results, and the certificates do not by themselves establish that the queue captures every reversible object of its length. A closing section sorts every claim in the paper by what a reader must accept in order to believe it, and states exactly how far the accompanying artifacts support each one.
ER -