AI 中文总结
本文通过调和 theta 恒等式和 Niemeier 构造，从秩26格重构广义六边形(2,8)，证明其唯一性，并利用射影3-设计第三矩证明紧致Cayley平面5-设计的几何唯一性，最终确定自同构群为 ^3D_4(2):3。
AI 中文摘要
我们从行列式为 $3$、最小范数为 $4$ 的抽象偶秩 $26$ 格重构阶为 $(2,8)$ 的广义六边形，反之亦然。调和 theta 恒等式确定了非零判别类中的 $819$ 个最短向量及其结合方案，并证明了该格及其对偶格的每个非空正范数壳层都是球面 $5$-设计。在反向方向上，秩 $26$ 幂等元给出该格，饱和性通过一个短的对偶陪集论证得到证明。一个正定 Niemeier 构造证明了该格的存在性和唯一性，从而也证明了六边形的存在性和唯一性。Cayley 平面中射影 $3$-设计的第三矩重构了无迹 Albert 积。这将保角双射提升为 $F_4(\mathbb R)$ 的元素，并证明了紧致 $819$ 点射影 $5$-设计的几何唯一性。最后，删除线给出了 $N(A_1^{24})$ 的一个无根指数四子格，具有 Golay 三元组和定向粘合。粘合数据形成八个根符号轨道，$L_3(2)\cong L_2(7)$ 在其上如同在 $\mathbf P^1(\mathbf{F}_7)$ 上作用；两个显式 Golay 置换证明了传递性。这证明了唯一性并给出了自同构群阶。该群随后被识别为 ${}^3D_4(2):3$，对于全格群有一个中心因子 $C_2$。重构和删除论证与长度 $26$ 二进制码构造平行。
英文摘要
We reconstruct the generalized hexagon of order $(2,8)$ from an abstract even rank-$26$ lattice of determinant $3$ and minimum $4$, and conversely reconstruct the lattice from the hexagon. Harmonic theta identities determine the $819$ shortest vectors in a non-zero discriminant class and their association scheme, and show that every non-empty positive-norm shell of the lattice and its dual is a spherical $5$-design. In the converse direction, the rank-$26$ idempotent gives the lattice, with saturation proved by a short dual-coset argument. A positive-definite Niemeier construction proves existence and uniqueness of the lattice and hence of the hexagon. The third moment of a projective $3$-design in the Cayley plane reconstructs the traceless Albert product. This upgrades angle-preserving bijections to elements of $F_4(\mathbb R)$ and proves geometric uniqueness of the tight $819$-point projective $5$-design. Finally, line deletion gives a rootless index-four sublattice of $N(A_1^{24})$ with a Golay trio and an oriented gluing. The gluing data form eight root-sign orbits, on which $L_3(2)\cong L_2(7)$ acts as on $\mathbf P^1(\mathbf{F}_7)$; two explicit Golay permutations prove transitivity. This proves uniqueness and gives the automorphism-group order. The group is identified afterward as ${}^3D_4(2):3$, with a central factor $C_2$ for the full lattice group. The reconstruction and deletion arguments parallel the length-$26$ binary code construction.
Comments39 pages; LaTeX
