matroid #9

n: 22
rank: 9
bases: 176,265
α(M): 4664/4007 ≈ 1.1640

status: superseded — held the GF(5) record until beaten
discovered by: Chris Eur
discovery date: 2026-08-05
AI use: yes — ChatGPT 5.6 Sol Ultra: Found by prompting Codex to look for variations of the double-free extension construction in small finite fields.
submitted by: Christopher Eur
submitted at: 2026-08-06 20:33:30 UTC

Description (matrix) GF(5)

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

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

Maximizing pair

(20, 21); basis counts at (i, j) = (20, 21):

	`j` ∈ `B`	`j` ∉ `B`
`i` ∈ `B`	27,500	80,140
`i` ∉ `B`	15,625	53,000

Notes

This is a (not free) coextension of the free extension of the self-direct-sum of the matroid of the complete graph on 5 vertices.