AI 中文总结
研究长度为333的勒让德对在特定子群作用下的情况，通过多种方法排除不符合条件的子群，证明这样的对仅在子群阶数\(|H|\leq 6\)时存在，限制了固定共同乘数对称性，未解决无限制存在问题。
AI 中文摘要
长度为333的勒让德对将产生一个668阶的哈达玛矩阵，这是哈达玛猜想目前未解决的最小阶数。我们研究了两个序列由共同子群\(H\leq(\mathbb Z/333\mathbb Z)^\times\)通过坐标乘法固定的结构化情况。证明这样的对仅在\(|H|\leq 6\)时存在。通过模3压缩将问题简化为108阶核后有30个子群，排除21个，包括所有阶至少为9的19个子群，最后通过分析排除9阶子群。结果仅限制了固定的共同乘数对称性，无限制的存在问题仍未解决。
英文摘要
A Legendre pair of length 333 would yield a Hadamard matrix of order 668, the smallest order presently unresolved by the Hadamard conjecture. We study the structured case in which both sequences are fixed by a common subgroup $H\leq(\mathbb Z/333\mathbb Z)^\times$ acting by coordinate multiplication. We prove that such a pair can exist only when $|H|\leq 6$. After a mod-3 compression reduces the problem to an order-108 kernel, there are exactly 30 subgroups. We exclude 21 of them, including all 19 subgroups of order at least 9. The final order-9 subgroup is eliminated analytically: its orbit structure restricts the 9-compressed entries to $\{\pm1,\pm17,\pm19,\pm35,\pm37\}$; the Legendre equations force a $+17,-17$ pair in one compressed sequence, and a single-shift autocorrelation bound then contradicts the required compressed correlation. The remaining exclusions use full-image compression, a row-sum congruence, exact meet-in-the-middle enumeration, and proof-carrying pseudo-Boolean encodings. The solver-assisted cases are accompanied by independently checked DRAT proofs or direct arithmetic certificates. The result constrains fixed common-multiplier symmetry only; the unrestricted existence problems remain open.
Comments12 pages, 3 tables
