matroid #4

n: 22
rank: 5
bases: 9,625
α(M): 8/7 ≈ 1.1429

status: current record for this ground-set size
submitted by: David Renshaw
submitted at: 2026-08-05 15:39:09 UTC

Description (matrix) GF(5)

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

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

Maximizing pair

(0, 1); basis counts at (i, j) = (0, 1):

	`j` ∈ `B`	`j` ∉ `B`
`i` ∈ `B`	500	625
`i` ∉ `B`	3,500	5,000

Attribution

The finite-field spike M_5^5 of Huh, Schroter, and Wang (Correlation bounds for fields and matroids, arXiv:1806.02675, Definition 19; discovered June 2018, no AI use reported in that record). Columns over GF(5): the tip e1, the vector e2+e3+e4+e5, and k*e1+e_m for m=2..5, k=1..p. Submitted from the literature.

Commentary

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