Matroid Correlation
Constants

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matroid #19 — 17-element, rank-10 matroid over GF(4)

n
17
rank
10
bases
1,816
α(M)
27/23 ≈ 1.1739
status
superseded — held the GF(4) record until beaten
discovered by
Yongxi (Aaron) Lin
discovery date
2026-08-09
AI use
yes — GPT-5.6 Sol: GPT 5.6 found this example after I ask it to focus on matroids with smaller ground sets.
submitted by
Yongxi (Aaron) Lin
submitted at
2026-08-09 18:40:08 UTC

Description (matrix) GF(4)

[[1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
 [0,1,0,1,1,0,0,0,0,0,0,0,0,0,0,1,0],
 [0,0,1,1,[0,1],0,0,0,0,0,0,0,0,0,0,[1,1],0],
 [0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],
 [0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,1,0],
 [0,0,0,0,0,0,0,1,1,[0,1],0,0,0,0,0,[1,1],0],
 [0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0],
 [0,0,0,0,0,0,0,0,0,0,0,1,0,1,1,1,0],
 [0,0,0,0,0,0,0,0,0,0,0,0,1,1,[0,1],[1,1],0],
 [0,0,0,[1,1],[0,1],0,0,0,[1,1],[0,1],0,0,0,[1,1],[0,1],0,1]]
show all 1,816 bases 
[[0,1,2,3,5,6,7,10,11,12],
 [0,1,2,4,5,6,7,10,11,12],
 [0,1,3,4,5,6,7,10,11,12],
 [0,2,3,4,5,6,7,10,11,12],
 [0,1,2,3,5,6,8,10,11,12],
 [0,1,2,4,5,6,8,10,11,12],
 [0,1,3,4,5,6,8,10,11,12],
 [0,2,3,4,5,6,8,10,11,12],
 [0,1,2,3,5,7,8,10,11,12],
 [0,1,2,4,5,7,8,10,11,12],
 [0,1,3,4,5,7,8,10,11,12],
 [0,2,3,4,5,7,8,10,11,12],
 [0,1,2,5,6,7,8,10,11,12],
 [0,1,3,5,6,7,8,10,11,12],
 [0,2,3,5,6,7,8,10,11,12],
 [0,1,4,5,6,7,8,10,11,12],
 [0,2,4,5,6,7,8,10,11,12],
 [0,3,4,5,6,7,8,10,11,12],
 [0,1,2,3,5,6,9,10,11,12],
 [0,1,2,4,5,6,9,10,11,12],
 [0,1,3,4,5,6,9,10,11,12],
 [0,2,3,4,5,6,9,10,11,12],
 [0,1,2,3,5,7,9,10,11,12],
 [0,1,2,4,5,7,9,10,11,12],
 [0,1,3,4,5,7,9,10,11,12],
 [0,2,3,4,5,7,9,10,11,12],
 [0,1,2,5,6,7,9,10,11,12],
 [0,1,3,5,6,7,9,10,11,12],
 [0,2,3,5,6,7,9,10,11,12],
 [0,1,4,5,6,7,9,10,11,12],
 [0,2,4,5,6,7,9,10,11,12],
 [0,3,4,5,6,7,9,10,11,12],
 [0,1,2,3,5,8,9,10,11,12],
 [0,1,2,4,5,8,9,10,11,12],
 [0,1,3,4,5,8,9,10,11,12],
 [0,2,3,4,5,8,9,10,11,12],
 [0,1,2,5,6,8,9,10,11,12],
 [0,1,3,5,6,8,9,10,11,12],
 [0,2,3,5,6,8,9,10,11,12],
 [0,1,4,5,6,8,9,10,11,12],
 [0,2,4,5,6,8,9,10,11,12],
 [0,3,4,5,6,8,9,10,11,12],
 [0,1,2,5,7,8,9,10,11,12],
 [0,1,3,5,7,8,9,10,11,12],
 [0,2,3,5,7,8,9,10,11,12],
 [0,1,4,5,7,8,9,10,11,12],
 [0,2,4,5,7,8,9,10,11,12],
 [0,3,4,5,7,8,9,10,11,12],
 [0,1,2,3,5,6,7,10,11,13],
 [0,1,2,4,5,6,7,10,11,13],
 [0,1,3,4,5,6,7,10,11,13],
 [0,2,3,4,5,6,7,10,11,13],
 [0,1,2,3,5,6,8,10,11,13],
 [0,1,2,4,5,6,8,10,11,13],
 [0,1,3,4,5,6,8,10,11,13],
 [0,2,3,4,5,6,8,10,11,13],
 [0,1,2,3,5,7,8,10,11,13],
 [0,1,2,4,5,7,8,10,11,13],
 [0,1,3,4,5,7,8,10,11,13],
 [0,2,3,4,5,7,8,10,11,13],
 [0,1,2,5,6,7,8,10,11,13],
 [0,1,3,5,6,7,8,10,11,13],
 [0,2,3,5,6,7,8,10,11,13],
 [0,1,4,5,6,7,8,10,11,13],
 [0,2,4,5,6,7,8,10,11,13],
 [0,3,4,5,6,7,8,10,11,13],
 [0,1,2,3,5,6,9,10,11,13],
 [0,1,2,4,5,6,9,10,11,13],
 [0,1,3,4,5,6,9,10,11,13],
 [0,2,3,4,5,6,9,10,11,13],
 [0,1,2,3,5,7,9,10,11,13],
 [0,1,2,4,5,7,9,10,11,13],
 [0,1,3,4,5,7,9,10,11,13],
 [0,2,3,4,5,7,9,10,11,13],
 [0,1,2,5,6,7,9,10,11,13],
 [0,1,3,5,6,7,9,10,11,13],
 [0,2,3,5,6,7,9,10,11,13],
 [0,1,4,5,6,7,9,10,11,13],
 [0,2,4,5,6,7,9,10,11,13],
 [0,3,4,5,6,7,9,10,11,13],
 [0,1,2,3,5,8,9,10,11,13],
 [0,1,2,4,5,8,9,10,11,13],
 [0,1,3,4,5,8,9,10,11,13],
 [0,2,3,4,5,8,9,10,11,13],
 [0,1,2,5,6,8,9,10,11,13],
 [0,1,3,5,6,8,9,10,11,13],
 [0,2,3,5,6,8,9,10,11,13],
 [0,1,4,5,6,8,9,10,11,13],
 [0,2,4,5,6,8,9,10,11,13],
 [0,3,4,5,6,8,9,10,11,13],
 [0,1,2,5,7,8,9,10,11,13],
 [0,1,3,5,7,8,9,10,11,13],
 [0,2,3,5,7,8,9,10,11,13],
 [0,1,4,5,7,8,9,10,11,13],
 [0,2,4,5,7,8,9,10,11,13],
 [0,3,4,5,7,8,9,10,11,13],
 [0,1,2,3,5,6,7,10,12,13],
 [0,1,2,4,5,6,7,10,12,13],
 [0,1,3,4,5,6,7,10,12,13],
 [0,2,3,4,5,6,7,10,12,13],
 [0,1,2,3,5,6,8,10,12,13],
 [0,1,2,4,5,6,8,10,12,13],
 [0,1,3,4,5,6,8,10,12,13],
 [0,2,3,4,5,6,8,10,12,13],
 [0,1,2,3,5,7,8,10,12,13],
 [0,1,2,4,5,7,8,10,12,13],
 [0,1,3,4,5,7,8,10,12,13],
 [0,2,3,4,5,7,8,10,12,13],
 [0,1,2,5,6,7,8,10,12,13],
 [0,1,3,5,6,7,8,10,12,13],
 [0,2,3,5,6,7,8,10,12,13],
 [0,1,4,5,6,7,8,10,12,13],
 [0,2,4,5,6,7,8,10,12,13],
 [0,3,4,5,6,7,8,10,12,13],
 [0,1,2,3,5,6,9,10,12,13],
 [0,1,2,4,5,6,9,10,12,13],
 [0,1,3,4,5,6,9,10,12,13],
 [0,2,3,4,5,6,9,10,12,13],
 [0,1,2,3,5,7,9,10,12,13],
 [0,1,2,4,5,7,9,10,12,13],
 [0,1,3,4,5,7,9,10,12,13],
 [0,2,3,4,5,7,9,10,12,13],
 [0,1,2,5,6,7,9,10,12,13],
 [0,1,3,5,6,7,9,10,12,13],
 [0,2,3,5,6,7,9,10,12,13],
 [0,1,4,5,6,7,9,10,12,13],
 [0,2,4,5,6,7,9,10,12,13],
 [0,3,4,5,6,7,9,10,12,13],
 [0,1,2,3,5,8,9,10,12,13],
 [0,1,2,4,5,8,9,10,12,13],
 [0,1,3,4,5,8,9,10,12,13],
 [0,2,3,4,5,8,9,10,12,13],
 [0,1,2,5,6,8,9,10,12,13],
 [0,1,3,5,6,8,9,10,12,13],
 [0,2,3,5,6,8,9,10,12,13],
 [0,1,4,5,6,8,9,10,12,13],
 [0,2,4,5,6,8,9,10,12,13],
 [0,3,4,5,6,8,9,10,12,13],
 [0,1,2,5,7,8,9,10,12,13],
 [0,1,3,5,7,8,9,10,12,13],
 [0,2,3,5,7,8,9,10,12,13],
 [0,1,4,5,7,8,9,10,12,13],
 [0,2,4,5,7,8,9,10,12,13],
 [0,3,4,5,7,8,9,10,12,13],
 [0,1,2,5,6,7,10,11,12,13],
 [0,1,3,5,6,7,10,11,12,13],
 [0,2,3,5,6,7,10,11,12,13],
 [0,1,4,5,6,7,10,11,12,13],
 [0,2,4,5,6,7,10,11,12,13],
 [0,3,4,5,6,7,10,11,12,13],
 [0,1,2,5,6,8,10,11,12,13],
 [0,1,3,5,6,8,10,11,12,13],
 [0,2,3,5,6,8,10,11,12,13],
 [0,1,4,5,6,8,10,11,12,13],
 [0,2,4,5,6,8,10,11,12,13],
 [0,3,4,5,6,8,10,11,12,13],
 [0,1,2,5,7,8,10,11,12,13],
 [0,1,3,5,7,8,10,11,12,13],
 [0,2,3,5,7,8,10,11,12,13],
 [0,1,4,5,7,8,10,11,12,13],
 [0,2,4,5,7,8,10,11,12,13],
 [0,3,4,5,7,8,10,11,12,13],
 [0,1,2,5,6,9,10,11,12,13],
 [0,1,3,5,6,9,10,11,12,13],
 [0,2,3,5,6,9,10,11,12,13],
 [0,1,4,5,6,9,10,11,12,13],
 [0,2,4,5,6,9,10,11,12,13],
 [0,3,4,5,6,9,10,11,12,13],
 [0,1,2,5,7,9,10,11,12,13],
 [0,1,3,5,7,9,10,11,12,13],
 [0,2,3,5,7,9,10,11,12,13],
 [0,1,4,5,7,9,10,11,12,13],
 [0,2,4,5,7,9,10,11,12,13],
 [0,3,4,5,7,9,10,11,12,13],
 [0,1,2,5,8,9,10,11,12,13],
 [0,1,3,5,8,9,10,11,12,13],
 [0,2,3,5,8,9,10,11,12,13],
 [0,1,4,5,8,9,10,11,12,13],
 [0,2,4,5,8,9,10,11,12,13],
 [0,3,4,5,8,9,10,11,12,13],
 [0,1,2,3,5,6,7,10,11,14],
 [0,1,2,4,5,6,7,10,11,14],
 [0,1,3,4,5,6,7,10,11,14],
 [0,2,3,4,5,6,7,10,11,14],
 [0,1,2,3,5,6,8,10,11,14],
 [0,1,2,4,5,6,8,10,11,14],
 [0,1,3,4,5,6,8,10,11,14],
 [0,2,3,4,5,6,8,10,11,14],
 [0,1,2,3,5,7,8,10,11,14],
 [0,1,2,4,5,7,8,10,11,14],
 [0,1,3,4,5,7,8,10,11,14],
 [0,2,3,4,5,7,8,10,11,14],
 [0,1,2,5,6,7,8,10,11,14],
 [0,1,3,5,6,7,8,10,11,14],
 [0,2,3,5,6,7,8,10,11,14],
 [0,1,4,5,6,7,8,10,11,14],
 [0,2,4,5,6,7,8,10,11,14],
 [0,3,4,5,6,7,8,10,11,14],
 [0,1,2,3,5,6,9,10,11,14],
 [0,1,2,4,5,6,9,10,11,14],
 [0,1,3,4,5,6,9,10,11,14],
 [0,2,3,4,5,6,9,10,11,14],
 [0,1,2,3,5,7,9,10,11,14],
 [0,1,2,4,5,7,9,10,11,14],
 [0,1,3,4,5,7,9,10,11,14],
 [0,2,3,4,5,7,9,10,11,14],
 [0,1,2,5,6,7,9,10,11,14],
 [0,1,3,5,6,7,9,10,11,14],
 [0,2,3,5,6,7,9,10,11,14],
 [0,1,4,5,6,7,9,10,11,14],
 [0,2,4,5,6,7,9,10,11,14],
 [0,3,4,5,6,7,9,10,11,14],
 [0,1,2,3,5,8,9,10,11,14],
 [0,1,2,4,5,8,9,10,11,14],
 [0,1,3,4,5,8,9,10,11,14],
 [0,2,3,4,5,8,9,10,11,14],
 [0,1,2,5,6,8,9,10,11,14],
 [0,1,3,5,6,8,9,10,11,14],
 [0,2,3,5,6,8,9,10,11,14],
 [0,1,4,5,6,8,9,10,11,14],
 [0,2,4,5,6,8,9,10,11,14],
 [0,3,4,5,6,8,9,10,11,14],
 [0,1,2,5,7,8,9,10,11,14],
 [0,1,3,5,7,8,9,10,11,14],
 [0,2,3,5,7,8,9,10,11,14],
 [0,1,4,5,7,8,9,10,11,14],
 [0,2,4,5,7,8,9,10,11,14],
 [0,3,4,5,7,8,9,10,11,14],
 [0,1,2,3,5,6,7,10,12,14],
 [0,1,2,4,5,6,7,10,12,14],
 [0,1,3,4,5,6,7,10,12,14],
 [0,2,3,4,5,6,7,10,12,14],
 [0,1,2,3,5,6,8,10,12,14],
 [0,1,2,4,5,6,8,10,12,14],
 [0,1,3,4,5,6,8,10,12,14],
 [0,2,3,4,5,6,8,10,12,14],
 [0,1,2,3,5,7,8,10,12,14],
 [0,1,2,4,5,7,8,10,12,14],
 [0,1,3,4,5,7,8,10,12,14],
 [0,2,3,4,5,7,8,10,12,14],
 [0,1,2,5,6,7,8,10,12,14],
 [0,1,3,5,6,7,8,10,12,14],
 [0,2,3,5,6,7,8,10,12,14],
 [0,1,4,5,6,7,8,10,12,14],
 [0,2,4,5,6,7,8,10,12,14],
 [0,3,4,5,6,7,8,10,12,14],
 [0,1,2,3,5,6,9,10,12,14],
 [0,1,2,4,5,6,9,10,12,14],
 [0,1,3,4,5,6,9,10,12,14],
 [0,2,3,4,5,6,9,10,12,14],
 [0,1,2,3,5,7,9,10,12,14],
 [0,1,2,4,5,7,9,10,12,14],
 [0,1,3,4,5,7,9,10,12,14],
 [0,2,3,4,5,7,9,10,12,14],
 [0,1,2,5,6,7,9,10,12,14],
 [0,1,3,5,6,7,9,10,12,14],
 [0,2,3,5,6,7,9,10,12,14],
 [0,1,4,5,6,7,9,10,12,14],
 [0,2,4,5,6,7,9,10,12,14],
 [0,3,4,5,6,7,9,10,12,14],
 [0,1,2,3,5,8,9,10,12,14],
 [0,1,2,4,5,8,9,10,12,14],
 [0,1,3,4,5,8,9,10,12,14],
 [0,2,3,4,5,8,9,10,12,14],
 [0,1,2,5,6,8,9,10,12,14],
 [0,1,3,5,6,8,9,10,12,14],
 [0,2,3,5,6,8,9,10,12,14],
 [0,1,4,5,6,8,9,10,12,14],
 [0,2,4,5,6,8,9,10,12,14],
 [0,3,4,5,6,8,9,10,12,14],
 [0,1,2,5,7,8,9,10,12,14],
 [0,1,3,5,7,8,9,10,12,14],
 [0,2,3,5,7,8,9,10,12,14],
 [0,1,4,5,7,8,9,10,12,14],
 [0,2,4,5,7,8,9,10,12,14],
 [0,3,4,5,7,8,9,10,12,14],
 [0,1,2,5,6,7,10,11,12,14],
 [0,1,3,5,6,7,10,11,12,14],
 [0,2,3,5,6,7,10,11,12,14],
 [0,1,4,5,6,7,10,11,12,14],
 [0,2,4,5,6,7,10,11,12,14],
 [0,3,4,5,6,7,10,11,12,14],
 [0,1,2,5,6,8,10,11,12,14],
 [0,1,3,5,6,8,10,11,12,14],
 [0,2,3,5,6,8,10,11,12,14],
 [0,1,4,5,6,8,10,11,12,14],
 [0,2,4,5,6,8,10,11,12,14],
 [0,3,4,5,6,8,10,11,12,14],
 [0,1,2,5,7,8,10,11,12,14],
 [0,1,3,5,7,8,10,11,12,14],
 [0,2,3,5,7,8,10,11,12,14],
 [0,1,4,5,7,8,10,11,12,14],
 [0,2,4,5,7,8,10,11,12,14],
 [0,3,4,5,7,8,10,11,12,14],
 [0,1,2,5,6,9,10,11,12,14],
 [0,1,3,5,6,9,10,11,12,14],
 [0,2,3,5,6,9,10,11,12,14],
 [0,1,4,5,6,9,10,11,12,14],
 [0,2,4,5,6,9,10,11,12,14],
 [0,3,4,5,6,9,10,11,12,14],
 [0,1,2,5,7,9,10,11,12,14],
 [0,1,3,5,7,9,10,11,12,14],
 [0,2,3,5,7,9,10,11,12,14],
 [0,1,4,5,7,9,10,11,12,14],
 [0,2,4,5,7,9,10,11,12,14],
 [0,3,4,5,7,9,10,11,12,14],
 [0,1,2,5,8,9,10,11,12,14],
 [0,1,3,5,8,9,10,11,12,14],
 [0,2,3,5,8,9,10,11,12,14],
 [0,1,4,5,8,9,10,11,12,14],
 [0,2,4,5,8,9,10,11,12,14],
 [0,3,4,5,8,9,10,11,12,14],
 [0,1,2,3,5,6,7,10,13,14],
 [0,1,2,4,5,6,7,10,13,14],
 [0,1,3,4,5,6,7,10,13,14],
 [0,2,3,4,5,6,7,10,13,14],
 [0,1,2,3,5,6,8,10,13,14],
 [0,1,2,4,5,6,8,10,13,14],
 [0,1,3,4,5,6,8,10,13,14],
 [0,2,3,4,5,6,8,10,13,14],
 [0,1,2,3,5,7,8,10,13,14],
 [0,1,2,4,5,7,8,10,13,14],
 [0,1,3,4,5,7,8,10,13,14],
 [0,2,3,4,5,7,8,10,13,14],
 [0,1,2,5,6,7,8,10,13,14],
 [0,1,3,5,6,7,8,10,13,14],
 [0,2,3,5,6,7,8,10,13,14],
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 [0,2,5,6,8,10,12,14,15,16],
 [0,3,5,6,8,10,12,14,15,16],
 [0,4,5,6,8,10,12,14,15,16],
 [0,1,5,7,8,10,12,14,15,16],
 [0,2,5,7,8,10,12,14,15,16],
 [0,3,5,7,8,10,12,14,15,16],
 [0,4,5,7,8,10,12,14,15,16],
 [0,1,2,5,9,10,12,14,15,16],
 [0,1,3,5,9,10,12,14,15,16],
 [0,2,3,5,9,10,12,14,15,16],
 [0,1,4,5,9,10,12,14,15,16],
 [0,2,4,5,9,10,12,14,15,16],
 [0,3,4,5,9,10,12,14,15,16],
 [0,1,5,6,9,10,12,14,15,16],
 [0,2,5,6,9,10,12,14,15,16],
 [0,3,5,6,9,10,12,14,15,16],
 [0,4,5,6,9,10,12,14,15,16],
 [0,1,5,7,9,10,12,14,15,16],
 [0,2,5,7,9,10,12,14,15,16],
 [0,3,5,7,9,10,12,14,15,16],
 [0,4,5,7,9,10,12,14,15,16],
 [0,1,5,8,9,10,12,14,15,16],
 [0,2,5,8,9,10,12,14,15,16],
 [0,3,5,8,9,10,12,14,15,16],
 [0,4,5,8,9,10,12,14,15,16],
 [0,1,2,5,6,10,13,14,15,16],
 [0,1,3,5,6,10,13,14,15,16],
 [0,2,3,5,6,10,13,14,15,16],
 [0,1,4,5,6,10,13,14,15,16],
 [0,2,4,5,6,10,13,14,15,16],
 [0,3,4,5,6,10,13,14,15,16],
 [0,1,2,5,7,10,13,14,15,16],
 [0,1,3,5,7,10,13,14,15,16],
 [0,2,3,5,7,10,13,14,15,16],
 [0,1,4,5,7,10,13,14,15,16],
 [0,2,4,5,7,10,13,14,15,16],
 [0,3,4,5,7,10,13,14,15,16],
 [0,1,5,6,7,10,13,14,15,16],
 [0,2,5,6,7,10,13,14,15,16],
 [0,3,5,6,7,10,13,14,15,16],
 [0,4,5,6,7,10,13,14,15,16],
 [0,1,2,5,8,10,13,14,15,16],
 [0,1,3,5,8,10,13,14,15,16],
 [0,2,3,5,8,10,13,14,15,16],
 [0,1,4,5,8,10,13,14,15,16],
 [0,2,4,5,8,10,13,14,15,16],
 [0,3,4,5,8,10,13,14,15,16],
 [0,1,5,6,8,10,13,14,15,16],
 [0,2,5,6,8,10,13,14,15,16],
 [0,3,5,6,8,10,13,14,15,16],
 [0,4,5,6,8,10,13,14,15,16],
 [0,1,5,7,8,10,13,14,15,16],
 [0,2,5,7,8,10,13,14,15,16],
 [0,3,5,7,8,10,13,14,15,16],
 [0,4,5,7,8,10,13,14,15,16],
 [0,1,2,5,9,10,13,14,15,16],
 [0,1,3,5,9,10,13,14,15,16],
 [0,2,3,5,9,10,13,14,15,16],
 [0,1,4,5,9,10,13,14,15,16],
 [0,2,4,5,9,10,13,14,15,16],
 [0,3,4,5,9,10,13,14,15,16],
 [0,1,5,6,9,10,13,14,15,16],
 [0,2,5,6,9,10,13,14,15,16],
 [0,3,5,6,9,10,13,14,15,16],
 [0,4,5,6,9,10,13,14,15,16],
 [0,1,5,7,9,10,13,14,15,16],
 [0,2,5,7,9,10,13,14,15,16],
 [0,3,5,7,9,10,13,14,15,16],
 [0,4,5,7,9,10,13,14,15,16],
 [0,1,5,8,9,10,13,14,15,16],
 [0,2,5,8,9,10,13,14,15,16],
 [0,3,5,8,9,10,13,14,15,16],
 [0,4,5,8,9,10,13,14,15,16]]

Maximizing pair

(15, 16); basis counts at (i, j) = (15, 16):

j ∈ Bj ∉ B
i ∈ B432736
i ∉ B216432

Notes

This matrix is of the form [ [C, 0, 0, u, 0], [0, C, 0, u, 0], [0, 0, C, u, 0], [h, h, h, 0, 1] ] where C = [ [1, 0, 0, 0, 0], [0, 1, 0, 1, 1], [0, 0, 1, 1, a] ] u = [ [0], [1], [a+1] ] h = [0, 0, 0, a+1, a]