matroid #14 (Three-block sparse affine-lift GF(4) record)

matroid #14 — Three-block sparse affine-lift GF(4) record

n: 25
rank: 13
bases: 661,932
α(M): 2915/2484 ≈ 1.1735

status: superseded — held the GF(4) record until beaten
discovered by: Jeewon Kim
discovery date: 2026-08-09
AI use: yes — GPT-5.6 Sol: AI performed the finite-geometry construction search, exact finite-field computations, and exhaustive all-bases/all-pairs verification. The original search produced the 26-element 27/23 example; a one-column deletion was subsequently checked to obtain this verifier-compatible 25-element record. (chat/run link)
submitted by: Jeewon Kim
submitted at: 2026-08-09 14:01:01 UTC

Description (matrix) GF(4)

[[0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
 [1,0,0,[1,1],1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0],
 [0,1,0,[1,1],1,0,[1,1],0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,[0,1],0],
 [0,0,1,[1,1],[1,1],[0,1],0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
 [0,0,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0],
 [0,0,0,0,0,0,0,0,1,0,0,[1,1],1,0,1,0,0,0,0,0,0,0,0,1,0],
 [0,0,0,0,0,0,0,0,0,1,0,[1,1],1,0,[1,1],0,0,0,0,0,0,0,0,[0,1],0],
 [0,0,0,0,0,0,0,0,0,0,1,[1,1],[1,1],[0,1],0,0,0,0,0,0,0,0,0,0,0],
 [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0],
 [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,[1,1],1,0,1,1,0],
 [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,[1,1],1,0,[1,1],[0,1],0],
 [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,[1,1],[1,1],[0,1],0,0,0],
 [0,1,[1,1],1,1,0,0,1,0,0,[1,1],0,[1,1],0,[0,1],[0,1],[0,1],1,1,0,1,0,[1,1],0,1]]

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

Maximizing pair

(23, 24); basis counts at (i, j) = (23, 24):

	`j` ∈ `B`	`j` ∉ `B`
`i` ∈ `B`	160,380	268,272
`i` ∉ `B`	78,732	154,548

Notes

Let GF(4) = {0,1,a,a+1}, where a^2+a+1=0. 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, 8-column seed C = [1 0 0 0 1 0 1 0 0 1 0 0 a+1 1 0 1 0 0 1 0 a+1 1 0 a+1 0 0 0 1 a+1 a+1 a 0], C^- is C with its first column deleted, u = [0,1,a,0]^T, h1 = [0,1,a+1,1,1,0,0], h2 = [1,0,0,a+1,0,a+1,0,a], h3 = [a,a,1,1,0,1,0,a+1]. The final two columns are the distinguished pair. The matroid has rank 13 and 25 elements. Exhaustive basis enumeration and an all-pairs check give the exact global correlation constant alpha-bar = 2915/2484, attained uniquely by the final two columns.