Sidon sets in F_3^7: a 34-point example and the upper bound 46

A set S ⊆ F3n is a 2-cap if no three of its points are collinear and no four are coplanar; equivalently, S is a Sidon set in the group (F3n, +). Let r(F3n) be the largest size of such a set. Huang, Tait and Won proved 33 ≤ r(F37) ≤ 47, the upper bound being the counting bound k(k−1) ≤ 3n−1. We show 34 ≤ r(F37) ≤ 46. The lower bound is an explicit 34-point set; we found seven pairwise affinely inequivalent examples, two of which are unions of orbits of linear maps of order 8. The upper bound combines the observation that the squared Fourier coefficients of a Sidon set are norms from Z[ω] with Delsarte's linear programming bound for the code spanned by the non-differences of S; the resulting certificate is short and can be checked by hand. We also record computational results: hyperplane sections of graphs of planar functions on F81 have at most 33 points; a 2-cap in F37 with an affine automorphism of order 7 has at most 30 points; and no 35-point 2-cap shares 27 or more points with any of four of our 34-point sets. This record contains the paper (PDF and HTML source), the seven 34-point sets and the other point sets used in the paper, the exact rational certificates for the upper bound (two independent Delsarte/Farkas certificates) and for the remarks on the linear programming method, stand-alone verification programs in pure Python (standard library only), and a Lean 4 / Mathlib proof that one of the 34-point sets is a Sidon set with no four coplanar points. See README.md for the commands and their expected output. Version 2 (2026-10-30) corrects Remark (iii) of the paper: for n = 5 the linear programming method gives only the counting bound 16, and 15 only under the extra assumption that the non-differences span F35. The results, point sets and certificates are unchanged. The preprint (version 1) was first posted on alphaXiv: https://www.alphaxiv.org/abs/2610.sidon-sets-34-point-example.