matroid #20 (Asymmetric 6-8-8 three-block affine lift over GF(4))

matroid #20 — Asymmetric 6-8-8 three-block affine lift over GF(4)

n: 24
rank: 13
bases: 160,704
α(M): 875/744 ≈ 1.1761

status: current GF(4) record
discovered by: Yongxi (Aaron) Lin
discovery date: 2026-08-09
AI use: yes — GPT-5.6 Sol: The user proposed truncating one block and allowing asymmetric blocks. ChatGPT derived the local block-count factorization, performed the symmetry-reduced finite-geometry search, and exhaustively enumerated all 160,704 bases and all 276 element pairs for the final matrix.
submitted by: Yongxi (Aaron) Lin
submitted at: 2026-08-09 18:56:27 UTC

Description (matrix) GF(4)

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

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

Maximizing pair

(22, 23); basis counts at (i, j) = (22, 23):

	`j` ∈ `B`	`j` ∉ `B`
`i` ∈ `B`	37,800	66,960
`i` ∉ `B`	18,144	37,800

Notes

Start from the idea of matroid #14 (three affine-lift blocks). Allow one block to have fewer columns. Search directly over the smaller block. The matrix is of the form [ C1 0 0 u1 0 ] [ 0 C2 0 u2 0 ] [ 0 0 C3 u3 0 ] [ h1 h2 h3 0 1 ] where C₁ = [ [1,0,0,0,0,0] [0,1,0,0,0,0] [0,0,1,0,1,1] [0,0,0,1,1,a] ] C₂ = [ [1,0,0,0,1,1,0,1] [0,1,0,0,0,1,1,a] [0,0,1,0,a,a,a,a] [0,0,0,1,0,1,a+1,a+1] ] C₃ = [ [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] ] u₁ = [ 0 0 a 1 ] u₂ = [ a+1 a 1 1 ] u₃ = [ 0 1 a 0 ] h₁ = [0,0,a,0,0,a+1] h₂ = [0,0,0,a+1,0,a,0,a+1] h₃ = [0,0,a,0,0,0,0,a]