matroid #18 (GF(5) three-block lift with two parallel replications)

matroid #18 — GF(5) three-block lift with two parallel replications

n: 25
rank: 13
bases: 560,422
α(M): 1045392/880397 ≈ 1.1874

status: current GF(5) record
discovered by: Yongxi (Aaron) Lin
discovery date: 2026-08-09
AI use: yes — GPT-5.6 Sol: ChatGPT started from the existing 23-element GF(5) three-block record construction (#15) and searched for improvements obtained by adding parallel copies of columns.
submitted by: Yongxi (Aaron) Lin
submitted at: 2026-08-09 18:34:38 UTC

Description (matrix) GF(5)

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

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

Maximizing pair

(21, 22); basis counts at (i, j) = (21, 22):

	`j` ∈ `B`	`j` ∉ `B`
`i` ∈ `B`	132,176	251,542
`i` ∉ `B`	54,208	122,496

Notes

This matroid is obtained from the construction in #15 by adding parallel copies of columns 2 and 9.