The affine geometry of rank n+1 over F_p is the matroid whose ground set consists of all vectors in a vector space over F_p of dimension n, where independence is given by affine independence, i.e. vectors are dependent iff there is a linear combination equaling zero in which the coefficients sum to zero (equivalently, the vectors are placed in the hyperplane x_0 = 1 in a vector space of dimension n+1, with ordinary linear independence in the larger space).
i1 : M = affineGeometry(3, 2) o1 = a matroid of rank 4 on 8 elements o1 : Matroid |
i2 : M === specificMatroid "AG32" o2 = true |
i3 : circuits M
o3 = {set {0, 1, 2, 3}, set {0, 1, 4, 5}, set {2, 3, 4, 5}, set {0, 2, 4, 6},
------------------------------------------------------------------------
set {1, 3, 4, 6}, set {1, 2, 5, 6}, set {0, 3, 5, 6}, set {1, 2, 4, 7},
------------------------------------------------------------------------
set {0, 3, 4, 7}, set {0, 2, 5, 7}, set {1, 3, 5, 7}, set {0, 1, 6, 7},
------------------------------------------------------------------------
set {2, 3, 6, 7}, set {4, 5, 6, 7}}
o3 : List
|
i4 : representationOf M
o4 = | 1 1 1 1 1 1 1 1 |
| 0 0 0 0 1 1 1 1 |
| 0 0 1 1 0 0 1 1 |
| 0 1 0 1 0 1 0 1 |
ZZ 4 ZZ 8
o4 : Matrix (--) <--- (--)
2 2
|
The object affineGeometry is a method function.