Legendre pairs of length 115 and the other open lengths below 200

Find a Legendre pair of length 115, 145, 159, 161, 169, 175, 177, 185, 187 or 195, the open odd lengths below 200.

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	1	+# Legendre pairs of length 115 and the other open lengths below 200
	2	+
	3	+A Legendre pair is two $\pm 1$ sequences whose periodic autocorrelations cancel almost perfectly, and each pair yields a Hadamard matrix. They are conjectured to exist for every odd length, but for ten odd lengths below 200, starting with 115, nobody has found one.
	4	+
	5	+## Problem
	6	+
	7	+Let $\ell$ be odd and $m = (\ell - 1)/2$. For $A = (a_0, \dots, a_{\ell-1}) \in \{-1, +1\}^\ell$, define the periodic autocorrelation
	8	+
	9	+$$\mathrm{PAF}_A(s) = \sum_{i=0}^{\ell-1} a_i\, a_{(i+s) \bmod \ell}.$$
	10	+
	11	+A Legendre pair of length $\ell$ is a pair $A, B \in \{-1, +1\}^\ell$ with
	12	+
	13	+$$\mathrm{PAF}_A(s) + \mathrm{PAF}_B(s) = -2 \quad (s = 1, \dots, m), \qquad \sum_i a_i = \sum_i b_i = 1.$$
	14	+
	15	+Because $\mathrm{PAF}(s) = \mathrm{PAF}(\ell - s)$, this covers every nonzero shift. Find a Legendre pair for one of the open lengths
	16	+
	17	+$$\ell \in \{115, 145, 159, 161, 169, 175, 177, 185, 187, 195\},$$
	18	+
	19	+or prove that none exists for one of them.
	20	+
	21	+## Background
	22	+
	23	+A Legendre pair of length $\ell$ gives, through a two-circulant-core construction, a Hadamard matrix of order $2\ell + 2$, so the conjecture that Legendre pairs exist for every odd $\ell$ is a structured route towards the Hadamard conjecture. The problem is problem 84 of CSPLib, proposed by Ilias Kotsireas, as a constraint satisfaction problem with $2\ell$ variables, $m$ quadratic constraints and two linear ones. Searches combine compression (summing entries over cosets), power spectral density tests and decimation symmetries, and have settled lengths one by one. The formal-conjectures project has a Lean statement of the Hadamard conjecture itself: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Wikipedia/Hadamard.lean
	24	+
	25	+## What is known
	26	+
	27	+- Kotsireas and Koutschan (2021) found pairs of lengths 117, 129, 133 and 147, and Kotsireas, Koutschan, Bulutoglu, Arquette, Turner and Ryan (2023) found lengths 85 and 87. That reduced the open lengths below 200 to the ten listed above.
	28	+- Kotsireas, Koutschan and Winterhof (2025) confirm that 115 is still the smallest unsettled length.
	29	+- At $\ell = 115$, two partial searches restricted by a multiplier symmetry have found nothing; the first was stopped after covering almost 4% of its search space, which took about 4,359 CPU days.
	30	+- Hadamard matrices of order $2\ell + 2$ are already known for these lengths by other constructions, so the question is specifically about Legendre pairs.
	31	+
	32	+## What would count as progress
	33	+
	34	+- A pair $(A, B)$ for one of the open lengths, given as two lists of $\pm 1$. A script checks all $m$ autocorrelation conditions and both sums in $O(\ell^2)$ time, a few milliseconds.
	35	+- A proof that no Legendre pair exists for some odd length, which would refute the conjecture; this would need a certificate or exhaustive search that others can reproduce, and is a separate and much harder kind of progress.
	36	+
	37	+## References
	38	+
	39	+1. I. Kotsireas and C. Koutschan (2021). Legendre pairs of lengths ℓ ≡ 0 (mod 3). Journal of Combinatorial Designs 29, 870–887. https://doi.org/10.1002/jcd.21806
	40	+2. I. Kotsireas, C. Koutschan, D. Bulutoglu, D. Arquette, J. Turner and K. Ryan (2023). Legendre pairs of lengths ℓ ≡ 0 (mod 5). Special Matrices 11, 20230105. https://doi.org/10.1515/spma-2023-0105
	41	+3. I. S. Kotsireas, C. Koutschan and A. Winterhof (2025). Quaternary Legendre pairs II. Discrete Mathematics 348, 114501. https://doi.org/10.1016/j.disc.2025.114501
	42	+4. CSPLib problem 84, 2cc Hadamard matrix Legendre pairs, with the CSP instances for odd ℓ up to 99. https://www.csplib.org/Problems/prob084/
	43	+
	44	+## Source
	45	+
	46	+Imported from [CSPLib prob084](https://www.csplib.org/Problems/prob084/), curated by CSPLib editors. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.