Matroid Correlation
Constants

 Submit a matroid

← all matroids

matroid #11 — Optimized version of #6

n
26
rank
9
bases
1,547,573
α(M)
98252/84369 ≈ 1.1646
status
superseded — held the GF(7) record until beaten
discovered by
Yongxi (Aaron) Lin
discovery date
2026-08
AI use
yes — GPT-5.6 Sol: AI found the pattern described in the notes completely by itself.
submitted by
Yongxi (Aaron) Lin
submitted at
2026-08-09 06:55:20 UTC

Description (matrix) GF(7)

[[5,3,2,4,5,3,2,3,3,3,2,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0],
 [1,0,1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
 [4,1,6,1,6,6,2,1,0,4,4,4,0,0,0,0,0,0,0,0,0,0,0,0,2,0],
 [5,2,2,3,0,1,3,1,4,1,4,2,0,0,0,0,0,0,0,0,0,0,0,0,1,0],
 [0,0,0,0,0,0,0,0,0,0,0,0,5,3,2,4,5,3,2,3,3,3,2,4,1,0],
 [0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,1,1,1,0,1,1,1,1,0,0],
 [0,0,0,0,0,0,0,0,0,0,0,0,4,1,6,1,6,6,2,1,0,4,4,4,2,0],
 [0,0,0,0,0,0,0,0,0,0,0,0,5,2,2,3,0,1,3,1,4,1,4,2,1,0],
 [0,0,0,0,1,2,6,0,2,3,5,1,0,0,0,0,6,5,1,0,5,4,2,6,0,1]]

The full basis list is too large to store; recompute it from the description above.

Maximizing pair

(24, 25); basis counts at (i, j) = (24, 25):

j ∈ Bj ∉ B
i ∈ B183,106506,214
i ∉ B203,401654,852

Notes

Write the matrix as [C 0 u 0; 0 C u 0; x y gamma 1]. Here C is a rank-4 matroid K on 12 elements, column 24 is represented by [u,u,gamma], and column 25 is represented by [0,0,1]. GPT claims that for this fixed C, this construction is optimized by taking u = [1,0,2,1], x = [0,0,0,0,1,2,6,0,2,3,5,1], y = -x, and gamma = 0.