An explicit prescribed Legendre pair of length 117
Abstract
A Legendre pair consists of two cyclic sign sequences whose periodic autocorrelations sum to −2 at every nonzero shift. An explicit pair of length 117 is constructed whose sums over equal residues modulo 13 are 1 at residue zero and respectively 3χ(r) and −3χ(r) elsewhere, where χ is the quadratic character modulo 13. This verifies one case of Kotsireas’s prescribed lifting conjecture, without establishing the general conjecture or classifying all lifts. Both sequences are fixed by an involution that reflects one Chinese remainder coordinate. A short correlation table proves the pair identities and gives a Hadamard matrix of order 236. The source archive contains the literal rows, manuscript and Lean sources, and parent and lift-search programs. The article and explanatory text are licensed under CC BY 4.0; original code and Lean sources are licensed under MIT. Third-party terms are retained as described in LICENSES.md. Use of generative AI. OpenAI's ChatGPT and Codex were used in developing the mathematics, writing and checking the programs, and preparing the text. The author is responsible for the mathematical statements, computations and references.
