[3321,369,81]₂. A weight-16 dual orbit has 13,450,050 checks, replication 64,800, and maximum pair codegree 810, so the dual-shell moment bound gives distance 81.The exact footprint-distance law d=q² is now certified at q=3,5,7,9. The q=5 apartment code remains the already-proved [73125,625,625]₂ code with exactly 936 chamber-star minima.
For odd q, writing g=q(q²+1)/2, the same-point chamber-star difference code has dimension q⁴-g. The footprint-rank equality rank₂(F)=g is equivalent to the concrete generation theorem K₀=D_q: every zero-P word is generated by same-point chamber-star differences. Exact rank equality now holds computationally at q=3,5,7,9,11; the all-odd theorem remains open.
The 13-cell cover stabilizer has order 576 and six orbits on triples of cells. Exact enumeration gives minimum 40 in every three-cell span. Thus a shortened word of weight 28, 32, or 36 must use at least four cells. The exact shortened distance is still open.
The q=5 point fiber is PG(3,2), with point-stabilizer action PSL₂(5) ≅ A₅. Its 35 projective lines split into orbits 5,10,10,10; the five-line orbit is a spread. The 2,340 minimum words of the q=5 zero-P residual span all 560 dimensions of that residual.
The q=5,7,9 low dual-support graphs share the finite skeleton K_(q−1,q−1) plus a 2-factor in each half. At q=9 the support graph is a Cayley graph on Z₈×Z₂.