matroid #6 (Double-extension sample over GF(7)) — Matroid Correlation Constants

matroid #6 — Double-extension sample over GF(7)

n: 26
rank: 9
bases: 1,518,400
α(M): 280043/243256 ≈ 1.1512

status: current record for this ground-set size
discovered by: Chris Eur
discovery date: 2026-08
AI use: yes — ChatGPT 5.6 Sol Ultra: Found by prompting it to look at the website and beat one of the listed challenges.
submitted by: David Renshaw
submitted at: 2026-08-05 17:26:29 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],
 [4,1,6,1,6,6,2,1,0,4,4,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
 [5,2,2,3,0,1,3,1,4,1,4,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
 [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],
 [0,0,0,0,0,0,0,0,0,0,0,0,4,1,6,1,6,6,2,1,0,4,4,4,0,0],
 [0,0,0,0,0,0,0,0,0,0,0,0,5,2,2,3,0,1,3,1,4,1,4,2,0,0],
 [1,3,6,0,5,1,1,0,5,4,4,4,5,3,6,3,4,5,0,6,2,2,2,1,3,1]]

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`	174,086	486,512
`i` ∉ `B`	203,401	654,401

Notes

Sample challenge submission by Chris Eur, August 2026: a double-(co)-extended self-direct-sum over GF(7) with alpha = 280043/243256 > 8/7, strictly beating the GF(7) field record of 8/7. Discovered by prompting ChatGPT 5.6 Sol Ultra to look at the website and beat one of the listed challenges. The maximizing pair (24, 25) is the free extension and free coextension of the construction. Submitted on Chris's behalf.

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