matroid #3
- n
- 12
- rank
- 6
- bases
- 684
- α(M)
- 100/81 ≈ 1.2346
Description (matrix) GF(3)
[[1,2,1,2,2,2,1,1,0,0,1,2], [0,0,2,0,0,0,0,0,2,1,1,1], [0,1,1,1,0,1,1,2,1,1,1,2], [1,1,1,2,2,1,1,2,1,1,0,1], [0,1,0,1,2,1,1,0,2,0,1,0], [0,1,1,1,1,0,2,0,0,1,0,0]]
show all 684 bases
[[0,1,2,3,4,5], [0,1,2,3,4,6], [0,1,2,4,5,6], [0,2,3,4,5,6], [1,2,3,4,5,6], [0,1,2,3,4,7], [0,1,2,3,5,7], [0,2,3,4,5,7], [1,2,3,4,5,7], [0,1,2,3,6,7], [0,1,2,4,6,7], [0,2,3,4,6,7], [1,2,3,4,6,7], [0,1,2,5,6,7], [0,2,3,5,6,7], [1,2,3,5,6,7], [0,2,4,5,6,7], [1,2,4,5,6,7], [0,1,2,3,4,8], [0,1,2,3,5,8], [0,1,2,4,5,8], [0,1,3,4,5,8], [0,1,2,3,6,8], [0,1,3,4,6,8], [0,2,3,4,6,8], [1,2,3,4,6,8], [0,1,2,5,6,8], [0,2,3,5,6,8], [1,2,3,5,6,8], [0,1,4,5,6,8], [0,2,4,5,6,8], [1,2,4,5,6,8], [0,3,4,5,6,8], [1,3,4,5,6,8], [0,1,2,4,7,8], [0,1,3,4,7,8], [0,2,3,4,7,8], [1,2,3,4,7,8], [0,1,2,5,7,8], [0,1,3,5,7,8], [0,2,3,5,7,8], [1,2,3,5,7,8], [0,2,4,5,7,8], [1,2,4,5,7,8], [0,3,4,5,7,8], [1,3,4,5,7,8], [0,1,2,6,7,8], [0,1,3,6,7,8], [0,2,3,6,7,8], [1,2,3,6,7,8], [0,1,4,6,7,8], [0,2,4,6,7,8], [1,2,4,6,7,8], [0,3,4,6,7,8], [1,3,4,6,7,8], [0,1,5,6,7,8], [0,3,5,6,7,8], [1,3,5,6,7,8], [0,4,5,6,7,8], [1,4,5,6,7,8], [0,1,2,3,4,9], [0,1,2,3,5,9], [0,1,3,4,5,9], [1,2,3,4,5,9], [0,1,2,3,6,9], [0,1,2,4,6,9], [0,1,3,4,6,9], [0,2,3,4,6,9], [0,1,2,5,6,9], [0,2,3,5,6,9], [1,2,3,5,6,9], [0,1,4,5,6,9], [1,2,4,5,6,9], [0,3,4,5,6,9], [1,3,4,5,6,9], [2,3,4,5,6,9], [0,1,2,3,7,9], [0,1,3,4,7,9], [0,2,3,4,7,9], [1,2,3,4,7,9], [0,1,3,5,7,9], [0,2,3,5,7,9], [0,3,4,5,7,9], [1,3,4,5,7,9], [2,3,4,5,7,9], [0,1,2,6,7,9], [0,1,3,6,7,9], [1,2,3,6,7,9], [0,1,4,6,7,9], [0,2,4,6,7,9], [1,2,4,6,7,9], [0,3,4,6,7,9], [1,3,4,6,7,9], [2,3,4,6,7,9], [0,1,5,6,7,9], [0,2,5,6,7,9], [0,3,5,6,7,9], [1,3,5,6,7,9], [2,3,5,6,7,9], [0,4,5,6,7,9], [1,4,5,6,7,9], [2,4,5,6,7,9], [0,1,2,3,8,9], [0,1,2,4,8,9], [0,1,3,4,8,9], [1,2,3,4,8,9], [0,1,2,5,8,9], [1,2,3,5,8,9], [0,1,4,5,8,9], [1,2,4,5,8,9], [1,3,4,5,8,9], [0,1,2,6,8,9], [0,2,3,6,8,9], [1,2,3,6,8,9], [0,1,4,6,8,9], [0,2,4,6,8,9], [1,2,4,6,8,9], [0,3,4,6,8,9], [1,3,4,6,8,9], [2,3,4,6,8,9], [0,2,5,6,8,9], [2,3,5,6,8,9], [0,4,5,6,8,9], [2,4,5,6,8,9], [3,4,5,6,8,9], [0,1,2,7,8,9], [0,1,3,7,8,9], [0,2,3,7,8,9], [1,2,3,7,8,9], [0,1,4,7,8,9], [0,2,4,7,8,9], [1,2,4,7,8,9], [0,3,4,7,8,9], [2,3,4,7,8,9], [0,1,5,7,8,9], [0,2,5,7,8,9], [1,3,5,7,8,9], [2,3,5,7,8,9], [0,4,5,7,8,9], [1,4,5,7,8,9], [2,4,5,7,8,9], [3,4,5,7,8,9], [0,1,6,7,8,9], [0,2,6,7,8,9], [0,3,6,7,8,9], [1,3,6,7,8,9], [2,3,6,7,8,9], [1,4,6,7,8,9], [2,4,6,7,8,9], [3,4,6,7,8,9], [0,5,6,7,8,9], [3,5,6,7,8,9], [4,5,6,7,8,9], [0,1,2,3,4,10], [0,1,2,3,5,10], [0,1,2,4,5,10], [0,1,3,4,5,10], [0,2,3,4,5,10], [1,2,3,4,5,10], [0,1,2,3,6,10], [0,1,3,4,6,10], [0,2,3,4,6,10], [0,1,2,5,6,10], [0,2,3,5,6,10], [1,2,3,5,6,10], [0,1,4,5,6,10], [0,2,4,5,6,10], [0,3,4,5,6,10], [1,3,4,5,6,10], [2,3,4,5,6,10], [0,1,2,4,7,10], [0,1,3,4,7,10], [1,2,3,4,7,10], [0,1,2,5,7,10], [0,1,3,5,7,10], [1,2,3,5,7,10], [0,2,4,5,7,10], [1,2,4,5,7,10], [0,3,4,5,7,10], [1,3,4,5,7,10], [2,3,4,5,7,10], [0,1,2,6,7,10], [0,1,3,6,7,10], [1,2,3,6,7,10], [0,1,4,6,7,10], [0,2,4,6,7,10], [0,3,4,6,7,10], [1,3,4,6,7,10], [2,3,4,6,7,10], [0,1,5,6,7,10], [0,2,5,6,7,10], [1,2,5,6,7,10], [0,3,5,6,7,10], [1,3,5,6,7,10], [2,3,5,6,7,10], [0,4,5,6,7,10], [1,4,5,6,7,10], [2,4,5,6,7,10], [0,1,3,4,8,10], [0,2,3,4,8,10], [1,2,3,4,8,10], [0,1,3,5,8,10], [0,2,3,5,8,10], [1,2,3,5,8,10], [0,1,4,5,8,10], [0,2,4,5,8,10], [1,2,4,5,8,10], [0,3,4,5,8,10], [1,3,4,5,8,10], [0,1,3,6,8,10], [0,2,3,6,8,10], [1,2,3,6,8,10], [1,3,4,6,8,10], [2,3,4,6,8,10], [0,1,5,6,8,10], [0,2,5,6,8,10], [1,2,5,6,8,10], [0,3,5,6,8,10], [1,3,5,6,8,10], [2,3,5,6,8,10], [1,4,5,6,8,10], [2,4,5,6,8,10], [3,4,5,6,8,10], [0,1,4,7,8,10], [0,2,4,7,8,10], [1,2,4,7,8,10], [0,3,4,7,8,10], [2,3,4,7,8,10], [0,1,5,7,8,10], [0,2,5,7,8,10], [1,2,5,7,8,10], [0,3,5,7,8,10], [2,3,5,7,8,10], [0,4,5,7,8,10], [1,4,5,7,8,10], [2,4,5,7,8,10], [3,4,5,7,8,10], [0,1,6,7,8,10], [0,2,6,7,8,10], [1,2,6,7,8,10], [0,3,6,7,8,10], [2,3,6,7,8,10], [1,4,6,7,8,10], [2,4,6,7,8,10], [3,4,6,7,8,10], [0,5,6,7,8,10], [1,5,6,7,8,10], [3,5,6,7,8,10], [4,5,6,7,8,10], [0,1,2,3,9,10], [0,1,2,4,9,10], [0,2,3,4,9,10], [0,1,2,5,9,10], [0,1,3,5,9,10], [0,2,3,5,9,10], [1,2,3,5,9,10], [0,1,4,5,9,10], [1,2,4,5,9,10], [0,3,4,5,9,10], [2,3,4,5,9,10], [0,1,2,6,9,10], [0,1,3,6,9,10], [0,1,4,6,9,10], [0,2,4,6,9,10], [0,3,4,6,9,10], [0,1,5,6,9,10], [0,2,5,6,9,10], [1,2,5,6,9,10], [0,3,5,6,9,10], [1,3,5,6,9,10], [0,4,5,6,9,10], [1,4,5,6,9,10], [2,4,5,6,9,10], [3,4,5,6,9,10], [0,1,2,7,9,10], [0,1,3,7,9,10], [1,2,3,7,9,10], [0,1,4,7,9,10], [0,2,4,7,9,10], [1,2,4,7,9,10], [0,3,4,7,9,10], [2,3,4,7,9,10], [0,1,5,7,9,10], [0,2,5,7,9,10], [0,3,5,7,9,10], [1,3,5,7,9,10], [2,3,5,7,9,10], [0,4,5,7,9,10], [1,4,5,7,9,10], [2,4,5,7,9,10], [3,4,5,7,9,10], [1,2,6,7,9,10], [1,3,6,7,9,10], [1,4,6,7,9,10], [2,4,6,7,9,10], [3,4,6,7,9,10], [1,5,6,7,9,10], [2,5,6,7,9,10], [3,5,6,7,9,10], [4,5,6,7,9,10], [0,1,3,8,9,10], [0,2,3,8,9,10], [1,2,3,8,9,10], [0,1,4,8,9,10], [0,2,4,8,9,10], [1,2,4,8,9,10], [0,3,4,8,9,10], [2,3,4,8,9,10], [0,1,5,8,9,10], [0,2,5,8,9,10], [1,2,5,8,9,10], [1,3,5,8,9,10], [2,3,5,8,9,10], [0,4,5,8,9,10], [1,4,5,8,9,10], [2,4,5,8,9,10], [3,4,5,8,9,10], [0,1,6,8,9,10], [0,2,6,8,9,10], [1,2,6,8,9,10], [0,3,6,8,9,10], [1,3,6,8,9,10], [1,4,6,8,9,10], [2,4,6,8,9,10], [3,4,6,8,9,10], [0,5,6,8,9,10], [2,5,6,8,9,10], [3,5,6,8,9,10], [4,5,6,8,9,10], [0,1,7,8,9,10], [0,2,7,8,9,10], [1,2,7,8,9,10], [0,3,7,8,9,10], [2,3,7,8,9,10], [0,5,7,8,9,10], [1,5,7,8,9,10], [2,5,7,8,9,10], [3,5,7,8,9,10], [1,6,7,8,9,10], [2,6,7,8,9,10], [3,6,7,8,9,10], [5,6,7,8,9,10], [0,1,2,3,4,11], [0,1,2,4,5,11], [0,1,3,4,5,11], [0,2,3,4,5,11], [0,1,2,4,6,11], [0,1,3,4,6,11], [0,2,3,4,6,11], [1,2,3,4,6,11], [0,1,4,5,6,11], [1,2,4,5,6,11], [0,3,4,5,6,11], [1,3,4,5,6,11], [2,3,4,5,6,11], [0,1,2,3,7,11], [0,1,2,4,7,11], [0,1,3,4,7,11], [1,2,3,4,7,11], [0,1,2,5,7,11], [0,1,3,5,7,11], [0,2,3,5,7,11], [0,2,4,5,7,11], [1,2,4,5,7,11], [0,3,4,5,7,11], [1,3,4,5,7,11], [2,3,4,5,7,11], [0,1,2,6,7,11], [0,1,3,6,7,11], [0,2,3,6,7,11], [1,2,3,6,7,11], [0,1,4,6,7,11], [0,2,4,6,7,11], [0,3,4,6,7,11], [1,3,4,6,7,11], [2,3,4,6,7,11], [0,1,5,6,7,11], [1,2,5,6,7,11], [0,3,5,6,7,11], [1,3,5,6,7,11], [2,3,5,6,7,11], [0,4,5,6,7,11], [1,4,5,6,7,11], [2,4,5,6,7,11], [0,1,2,3,8,11], [0,1,2,4,8,11], [0,1,3,4,8,11], [0,2,3,4,8,11], [0,1,2,5,8,11], [0,1,3,5,8,11], [0,2,3,5,8,11], [0,2,4,5,8,11], [0,3,4,5,8,11], [0,1,2,6,8,11], [0,1,3,6,8,11], [0,2,3,6,8,11], [1,2,3,6,8,11], [0,1,4,6,8,11], [0,2,4,6,8,11], [1,2,4,6,8,11], [1,3,4,6,8,11], [2,3,4,6,8,11], [0,1,5,6,8,11], [1,2,5,6,8,11], [0,3,5,6,8,11], [1,3,5,6,8,11], [2,3,5,6,8,11], [0,4,5,6,8,11], [2,4,5,6,8,11], [3,4,5,6,8,11], [0,1,2,7,8,11], [0,1,3,7,8,11], [0,2,3,7,8,11], [1,2,3,7,8,11], [0,2,4,7,8,11], [1,2,4,7,8,11], [0,3,4,7,8,11], [1,3,4,7,8,11], [2,3,4,7,8,11], [1,2,5,7,8,11], [1,3,5,7,8,11], [2,3,5,7,8,11], [2,4,5,7,8,11], [3,4,5,7,8,11], [0,1,6,7,8,11], [1,2,6,7,8,11], [0,3,6,7,8,11], [2,3,6,7,8,11], [0,4,6,7,8,11], [1,4,6,7,8,11], [2,4,6,7,8,11], [3,4,6,7,8,11], [1,5,6,7,8,11], [3,5,6,7,8,11], [4,5,6,7,8,11], [0,1,2,3,9,11], [0,1,2,4,9,11], [0,2,3,4,9,11], [1,2,3,4,9,11], [0,1,2,5,9,11], [0,1,3,5,9,11], [0,2,3,5,9,11], [0,1,4,5,9,11], [1,2,4,5,9,11], [0,3,4,5,9,11], [1,3,4,5,9,11], [2,3,4,5,9,11], [0,1,2,6,9,11], [0,1,3,6,9,11], [0,2,3,6,9,11], [1,2,3,6,9,11], [0,1,4,6,9,11], [1,2,4,6,9,11], [0,3,4,6,9,11], [1,3,4,6,9,11], [2,3,4,6,9,11], [0,1,5,6,9,11], [1,2,5,6,9,11], [0,3,5,6,9,11], [1,3,5,6,9,11], [2,3,5,6,9,11], [0,1,2,7,9,11], [0,1,3,7,9,11], [0,2,3,7,9,11], [0,1,4,7,9,11], [0,2,4,7,9,11], [1,2,4,7,9,11], [0,3,4,7,9,11], [1,3,4,7,9,11], [2,3,4,7,9,11], [0,1,5,7,9,11], [0,2,5,7,9,11], [0,4,5,7,9,11], [1,4,5,7,9,11], [2,4,5,7,9,11], [0,2,6,7,9,11], [1,2,6,7,9,11], [0,3,6,7,9,11], [1,3,6,7,9,11], [2,3,6,7,9,11], [0,4,6,7,9,11], [1,4,6,7,9,11], [2,4,6,7,9,11], [0,5,6,7,9,11], [1,5,6,7,9,11], [2,5,6,7,9,11], [0,1,3,8,9,11], [0,2,3,8,9,11], [1,2,3,8,9,11], [0,1,4,8,9,11], [0,2,4,8,9,11], [1,2,4,8,9,11], [0,3,4,8,9,11], [1,3,4,8,9,11], [2,3,4,8,9,11], [0,1,5,8,9,11], [0,2,5,8,9,11], [1,2,5,8,9,11], [1,3,5,8,9,11], [2,3,5,8,9,11], [0,4,5,8,9,11], [2,4,5,8,9,11], [3,4,5,8,9,11], [0,1,6,8,9,11], [0,2,6,8,9,11], [1,2,6,8,9,11], [0,3,6,8,9,11], [1,3,6,8,9,11], [0,4,6,8,9,11], [2,4,6,8,9,11], [3,4,6,8,9,11], [0,5,6,8,9,11], [2,5,6,8,9,11], [3,5,6,8,9,11], [0,1,7,8,9,11], [0,2,7,8,9,11], [1,2,7,8,9,11], [1,3,7,8,9,11], [2,3,7,8,9,11], [0,4,7,8,9,11], [1,4,7,8,9,11], [3,4,7,8,9,11], [1,5,7,8,9,11], [2,5,7,8,9,11], [4,5,7,8,9,11], [0,6,7,8,9,11], [1,6,7,8,9,11], [2,6,7,8,9,11], [3,6,7,8,9,11], [4,6,7,8,9,11], [5,6,7,8,9,11], [0,1,2,3,10,11], [0,1,2,4,10,11], [1,2,3,4,10,11], [0,1,2,5,10,11], [0,1,3,5,10,11], [0,2,3,5,10,11], [0,1,4,5,10,11], [0,2,4,5,10,11], [1,2,4,5,10,11], [1,3,4,5,10,11], [2,3,4,5,10,11], [0,1,2,6,10,11], [0,1,3,6,10,11], [0,2,3,6,10,11], [1,2,3,6,10,11], [0,1,4,6,10,11], [0,2,4,6,10,11], [1,3,4,6,10,11], [2,3,4,6,10,11], [0,1,5,6,10,11], [1,2,5,6,10,11], [0,3,5,6,10,11], [1,3,5,6,10,11], [2,3,5,6,10,11], [0,4,5,6,10,11], [1,4,5,6,10,11], [2,4,5,6,10,11], [3,4,5,6,10,11], [0,1,2,7,10,11], [0,1,3,7,10,11], [1,2,3,7,10,11], [0,1,4,7,10,11], [1,3,4,7,10,11], [0,1,5,7,10,11], [0,2,5,7,10,11], [1,2,5,7,10,11], [0,3,5,7,10,11], [1,3,5,7,10,11], [2,3,5,7,10,11], [0,4,5,7,10,11], [1,4,5,7,10,11], [3,4,5,7,10,11], [0,2,6,7,10,11], [0,3,6,7,10,11], [2,3,6,7,10,11], [0,4,6,7,10,11], [3,4,6,7,10,11], [0,5,6,7,10,11], [2,5,6,7,10,11], [3,5,6,7,10,11], [4,5,6,7,10,11], [0,1,3,8,10,11], [0,2,3,8,10,11], [1,2,3,8,10,11], [0,1,4,8,10,11], [0,2,4,8,10,11], [1,2,4,8,10,11], [1,3,4,8,10,11], [2,3,4,8,10,11], [0,1,5,8,10,11], [0,2,5,8,10,11], [1,2,5,8,10,11], [0,3,5,8,10,11], [1,3,5,8,10,11], [2,3,5,8,10,11], [0,4,5,8,10,11], [2,4,5,8,10,11], [3,4,5,8,10,11], [0,1,6,8,10,11], [0,2,6,8,10,11], [1,2,6,8,10,11], [1,4,6,8,10,11], [2,4,6,8,10,11], [0,5,6,8,10,11], [1,5,6,8,10,11], [2,5,6,8,10,11], [4,5,6,8,10,11], [0,1,7,8,10,11], [0,2,7,8,10,11], [1,2,7,8,10,11], [0,3,7,8,10,11], [2,3,7,8,10,11], [0,4,7,8,10,11], [1,4,7,8,10,11], [3,4,7,8,10,11], [1,5,7,8,10,11], [2,5,7,8,10,11], [3,5,7,8,10,11], [4,5,7,8,10,11], [0,6,7,8,10,11], [2,6,7,8,10,11], [4,6,7,8,10,11], [5,6,7,8,10,11], [0,2,3,9,10,11], [1,2,3,9,10,11], [0,2,4,9,10,11], [1,2,4,9,10,11], [2,3,4,9,10,11], [0,2,5,9,10,11], [1,2,5,9,10,11], [0,3,5,9,10,11], [1,3,5,9,10,11], [2,3,5,9,10,11], [0,4,5,9,10,11], [1,4,5,9,10,11], [2,4,5,9,10,11], [3,4,5,9,10,11], [0,2,6,9,10,11], [1,2,6,9,10,11], [0,3,6,9,10,11], [1,3,6,9,10,11], [0,4,6,9,10,11], [1,4,6,9,10,11], [2,4,6,9,10,11], [3,4,6,9,10,11], [0,5,6,9,10,11], [1,5,6,9,10,11], [2,5,6,9,10,11], [3,5,6,9,10,11], [0,2,7,9,10,11], [1,2,7,9,10,11], [0,3,7,9,10,11], [1,3,7,9,10,11], [2,3,7,9,10,11], [0,4,7,9,10,11], [1,4,7,9,10,11], [3,4,7,9,10,11], [0,5,7,9,10,11], [1,5,7,9,10,11], [2,5,7,9,10,11], [4,5,7,9,10,11], [2,6,7,9,10,11], [3,6,7,9,10,11], [4,6,7,9,10,11], [5,6,7,9,10,11], [0,3,8,9,10,11], [1,3,8,9,10,11], [0,4,8,9,10,11], [1,4,8,9,10,11], [3,4,8,9,10,11], [0,5,8,9,10,11], [1,5,8,9,10,11], [3,5,8,9,10,11], [4,5,8,9,10,11], [0,6,8,9,10,11], [1,6,8,9,10,11], [4,6,8,9,10,11], [5,6,8,9,10,11], [0,7,8,9,10,11], [1,7,8,9,10,11], [3,7,8,9,10,11], [5,7,8,9,10,11], [6,7,8,9,10,11]]
Maximizing pair
(5, 10); basis counts at (i, j) = (5, 10):
| j ∈ B | j ∉ B | |
|---|---|---|
| i ∈ B | 180 | 162 |
| i ∉ B | 162 | 180 |
Attribution
Found independently in two searches, August 2026: Chris Eur and David Renshaw via AlphaEvolve, and June Huh via the study of Mathieu groups with ChatGPT 5.5 Pro assisting. Huh observed that contracting two elements yields the matroid of the Steiner system S(3,4,10), related to duads and synthemes. Submitted from the discovery record.
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