Matroid Correlation
Constants

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matroid #16 — Three-block sparse affine-lift GF(7) record

n
23
rank
13
bases
77,827
α(M)
40500/33827 ≈ 1.1973
status
superseded — held the GF(7) record until beaten
discovered by
Jeewon Kim
discovery date
2026-08-09
AI use
yes — GPT-5.6 Sol Ultra: AI performed the computational construction search, optimized the finite-field block and affine-lift parameters, and exhaustively verified every basis and every eligible element pair. Two independent implementations reproduced the exact correlation constant. (chat/run link)
submitted by
Jeewon Kim
submitted at
2026-08-09 14:07:17 UTC

Description (matrix) GF(7)

[[1,0,0,0,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0],
 [2,1,0,0,5,1,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0],
 [2,0,1,0,4,3,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0],
 [3,0,0,1,3,3,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0],
 [0,0,0,0,0,0,0,1,0,0,0,1,1,1,0,0,0,0,0,0,0,1,0],
 [0,0,0,0,0,0,0,2,1,0,0,5,1,6,0,0,0,0,0,0,0,3,0],
 [0,0,0,0,0,0,0,2,0,1,0,4,3,1,0,0,0,0,0,0,0,6,0],
 [0,0,0,0,0,0,0,3,0,0,1,3,3,3,0,0,0,0,0,0,0,3,0],
 [0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,1,1,1,0],
 [0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,1,0,0,5,1,6,3,0],
 [0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,1,0,4,3,1,6,0],
 [0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,3,3,3,3,0],
 [2,1,6,0,1,1,4,6,0,6,6,2,6,4,5,2,5,2,2,0,4,0,1]]

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 ∈ Bj ∉ B
i ∈ B18,00033,827
i ∉ B8,00018,000

Notes

The construction has the block form [C 0 0 u 0 0 C 0 u 0 0 0 C u 0 h1 h2 h3 0 1], where C is the rank-4, 7-column seed C = [1 0 0 0 1 1 1 2 1 0 0 5 1 6 2 0 1 0 4 3 1 3 0 0 1 3 3 3], u = [1,3,6,3]^T, h1 = [2,1,6,0,1,1,4], h2 = [6,0,6,6,2,6,4], h3 = [5,2,5,2,2,0,4]. The final two columns are the distinguished pair. The matroid has rank 13 and 23 elements. Exhaustive basis enumeration and an all-pairs check give the exact global correlation constant alpha-bar = 40500/33827, attained by the final two columns.