matroid #12 (A variant of #11) — Matroid Correlation Constants

matroid #12 — A variant of #11

n: 26
rank: 9
bases: 1,478,356
α(M): 147888/125779 ≈ 1.1758

status: superseded — held the GF(7) record until beaten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 did the intensive computations.
submitted by: Yongxi (Aaron) Lin
submitted at: 2026-08-09 08:00:25 UTC

Description (matrix) GF(7)

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

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` ∈ `B`	`j` ∉ `B`
`i` ∈ `B`	178,880	503,116
`i` ∉ `B`	184,900	611,460

Notes

Write the matrix as [C 0 u 0; 0 C u 0; h -h 0 1], which also appears in #11. GPT claims for this new C, this construction is optimized by taking u=[1,3,6,2], h=[0,0,0,0,1,3,5,3,2,6,0,2].