from sage.matroids.advanced import *

### 2r elements:

#Tipless rank-r free spike.
#When r=3, is isomorphic to U_{3,6}; when r=4, is the unique tightening of the Vamos matroid.
def freeSpike(r):
    if r == 3:
        return matroids.Uniform(3,6)
    elif r < 3:
        raise AttributeError("Tipless free spike must have rank at least three.")

    E = range(1,2*r+1)
    circs = [[2*i+1,2*i+2,2*j+1,2*j+2] for i in range(r) for j in range(i+1,r)]
    CC = {
        3: circs,
        r: [E]
    }
    spike = Matroid(groundset=E, circuit_closures=CC)
    spike.rename('(Tipless) rank-' + str(r) + ' free spike: ' + repr(spike))
    return spike

#Tipped rank-r free spike.
def tippedFreeSpike(r):
    if r == 3:
        return tippedFree3spike()
    elif r < 3:
        raise AttributeError("Tipped free spike must have rank at least three.")

    E = range(2*r+1)
    tris = [[0,2*i+1,2*i+2] for i in range(r)]
    planes = [[2*i+1,2*i+2,2*j+1,2*j+2] for i in range(r) for j in range(i+1,r)]
    CC = {
        2: triangles,
        3: planes,
        r: [E]
    }
    spike = Matroid(groundset=E, circuit_closures=CC)
    spike.rename('Tipped rank-' + str(r) + ' free spike: ' + repr(spike))
    return spike

### 7 elements:

#An excluded minor for 2-regular matroids.  UPF is K_2.
def relaxedNonFano():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [1, 1, 0, 1],
        [1, 0, 1, 1],
        [0, 1, w, 1]
    ])
    relaxedNonFano = QuaternaryMatroid(reduced_matrix=A)
    relaxedNonFano.rename('F7=: ' + repr(relaxedNonFano))
    return relaxedNonFano

#Unique 3-connected extension of U_{3,6}.  Stabilizer for K_2.
def tippedFree3spike():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [1,   1,   1, 1],
        [1, w+1,   0, w],
        [1,   0, w+1, w]
    ])
    spike = QuaternaryMatroid(reduced_matrix=A, groundset=[0,3,5,1,4,6,2])
    spike.rename('Tipped rank-3 free spike: ' + repr(spike))
    return spike

### 8 elements:

#The matroid obtained from a AG(2,3)\e by a single delta-Y exchange on a triangle.
#An excluded minor for near-regular matroids.  UPF is S.
def AG23minusDY():
    A = Matrix(GF(3), [
        [1, 1, 1, 1],
        [1, 0, 1, 2],
        [2, 0, 1, 2],
        [2, 1, 1, 0]
    ])
    dy = TernaryMatroid(reduced_matrix=A)
    dy.rename('Delta-Y of AG(2,3)\e: ' + repr(dy))
    return dy

#An excluded minor for 2-regular matroids.  UPF is K_2.  Self-dual.
def TQ8():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [0,   w, 1,   1],
        [1,   0, w, w+1],
        [1,   w, 0,   w],
        [1, w+1, 1,   0]
    ])
    tp8 = QuaternaryMatroid(reduced_matrix=A, groundset=[1,7,5,3,8,6,4,2])
    tp8.rename('TQ8: ' + repr(tp8))
    return tp8

#P8-: obtained by relaxing one of the disjoint circuit-hyperplanes of P8.
#An excluded minor for 2-regular matroids.  UPF is K_2.  Self-dual.
def P8p():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [1,   1, 1, w],
        [1, w+1, 1, 0],
        [1,   0, w, w],
        [0,   1, 1, 1]
    ])
    p8p = QuaternaryMatroid(reduced_matrix=A, groundset=['a','c','b','f','d','e','g','h'])
    p8p.rename('P8-: ' + repr(p8p))
    return p8p

#An excluded minor for K_2-representable matroids.  UPF is G.  Self-dual.
#Uniquely GF(5)-representable.  (An excluded minor for H_2-representable matroids.)
def KP8():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [0, 1,   1,   1],
        [1, 0,   w,   w],
        [1, 1,   1, 1+w],
        [1, 1, 1+w,   0]
    ])
    kp8 = QuaternaryMatroid(reduced_matrix=A, groundset=[1,4,3,5,6,7,0,2])
    kp8.rename('KP8: ' + repr(kp8))
    return kp8

#An excluded minor for G- and K_2-representable matroids.  UPF is U_1^(2).  Self-dual.  (Also an excluded minor for H_2- and GF(5)-representable matroids.)
def Sp8():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [1, 1, w+1,   0],
        [1, 1,   0, w+1],
        [1, 0,   w,   w],
        [0, 1,   1,   1]
    ])
    sp = QuaternaryMatroid(reduced_matrix=A, groundset=[1,2,3,5,4,6,7,8])
    sp.rename('Sp8: ' + repr(sp))
    return sp

#An excluded minor for G- and K_2-representable matroids.
#UPF is (GF(2)(a,b),<a,b,a+1,b+1,ab+a+b>).  Self-dual.
#(Also an excluded minor for H_2- and GF(5)-representable matroids.)
def Sp8pp():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [1, w, 1, 0],
        [1, 1, 1, 1],
        [w, 0, 1, w],
        [0, w, 1, 1]
    ])
    sp = QuaternaryMatroid(reduced_matrix=A, groundset=[1,5,6,7,2,3,4,8])
    sp.rename('Sp8=: ' + repr(sp))
    return sp

#An excluded minor for G- and K_2-representable matroids.  Self-dual.  UPF is W.
#(Also an excluded minor for H_2- and GF(5)-representable matroids.)
def LP8():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [  1, 1,   1, 1],
        [w+1, w,   0, 1],
        [  1, 0, w+1, 1],
        [  0, w,   w, 1]
    ])
    lp8 = QuaternaryMatroid(reduced_matrix=A, groundset=['a','b','d','e','c','f','g','h'])
    lp8.rename('LP8: ' + repr(lp8))
    return lp8

#An excluded minor for G-, K_2-, H_4-, and GF(5)-representable matroids.  Self-dual.
#UPF is (Z[zeta,a],<zeta,a-zeta>) where zeta is solution to x^2-x+1 = 0 and a is an indeterminate.
def WQ8():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [1, 0, 1, w+1],
        [1, 1, 1,   1],
        [w, 1, 1,   0],
        [0, w, 1,   1]
    ])
    wq8 = QuaternaryMatroid(reduced_matrix=A, groundset=[0,1,3,4,2,5,6,7])
    wq8.rename('WQ8: ' + repr(wq8))
    return wq8

### 9 elements:

#An excluded minor for K_2-representable matroids, and a restriction of the Betsy Ross matroid.
#The UPF is G.
#Uniquely GF(5)-representable.  (An excluded minor for H_2-representable matroids.)
def BB9():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [  1, 0,   1, 1, 1, 1],
        [  0, 1,   w, 1, 0, w],
        [w+1, 1, w+1, 1, w, 0]
    ])
    bb9 = QuaternaryMatroid(reduced_matrix=A, groundset=['i','b','d','j','h','f','c','a','k'])
    bb9.rename('BB9: ' + repr(bb9))
    return bb9

#An excluded minor for K_2-representable matroids, and a single-element extension of TQ8.
#The UPF is G.
#Uniquely GF(5)-representable.  (An excluded minor for H_2-representable matroids.)
def TQ9():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [  1, 0, w, 1,   1],
        [w+1, 0, 0, w,   1],
        [  1, w, 0, 0, w+1],
        [  1, 1, 1, 1,   0]
    ])
    tq9 = QuaternaryMatroid(reduced_matrix=A, groundset=[1,4,6,0,2,5,3,7,8])
    tq9.rename('TQ9: ' + repr(tq9))
    return tq9

#An excluded minor for G- and K_2-representable matroids, and a single-element extension of TQ8.
#UPF is U_1^(2).
#(An excluded minor for H_2- and GF(5)-representable matroids.)
def TQ9p():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [  1, 1,   1, w+1,   1],
        [w+1, w, w+1, w+1, w+1],
        [  w, 0,   w,   1, w+1],
        [  0, 1, w+1, w+1,   0]
    ])
    tq9p = QuaternaryMatroid(reduced_matrix=A, groundset=[1,4,7,8,0,6,5,2,3])
    tq9p.rename('TQ9\': ' + repr(tq9p))
    return tq9p

#An excluded minor for K_2-representable matroids.  A Y-delta exchange on the unique triad gives A_9.
#The UPF is P_4.
def M8591():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [1, 1, 0, w,   1],
        [0, 1, 1, w, w+1],
        [1, 0, w, w,   1],
        [0, 0, 1, 1,   0]
    ])
    m8591 = QuaternaryMatroid(reduced_matrix=A)
    m8591.rename('M8591: ' + repr(m8591))
    return m8591

#An excluded minor for K_2-representable matroids.  A single-element extension of P_8-.
#The UPF is P_4.  Has a P8- minor (delete z).
#Uniquely GF(5)-representable.  (An excluded minor for H_2-representable matroids.)
def PP9():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [1,   1, 1, w, w],
        [1, 1+w, 1, 0, w],
        [1,   0, w, w, w],
        [0,   1, 1, 1, 1]
    ])
    pp9 = QuaternaryMatroid(reduced_matrix=A, groundset=['a','c','b','f','d','e','g','h','z'])
    pp9.rename('PP9: ' + repr(pp9))
    return pp9

#An excluded minor for K_2-representable matroids.  The UPF is G.
#In a DY*-equivalence class of 4 matroids,
# one of which can be obtained from BB9 by a segment-cosegment exchange on {a,d,i,j}.
#Uniquely GF(5)-representable.  (An excluded minor for H_2-representable matroids.)
def BB9gDY():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [  w, w, w+1, 1],
        [  w, 1,   0, 0],
        [w+1, 1,   0, 0],
        [  1, w, w+1, w],
        [  w, 0,   1, 1]
    ])
    m9573 = QuaternaryMatroid(reduced_matrix=A, groundset=['c','d','i','f','h','a','j','k','b'])
    m9573.rename('Segment cosegment exchange on BB9: ' + repr(m9573))
    return m9573

#An excluded minor for K_2-representable matroids.
#The UPF is P_4.
#Uniquely GF(5)-representable.  (An excluded minor for H_2-representable matroids.)
def A9():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [w+1, 1, w,   w, w, w],
        [  0, 1, 1, w+1, 0, w],
        [  w, 0, 1, w+1, w, 1]
    ])
    a9 = QuaternaryMatroid(reduced_matrix=A, groundset=[6,5,4,1,2,3,7,8,0])
    a9.rename('A9: ' + repr(a9))
    return a9

#An excluded minor for G- and K_2-representable matroids.
#In a DY*-equivalence class of 10 matroids.  UPF is U_1^(2).
#(An excluded minor for H_2- and GF(5)-representable matroids.)
def FN9():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [w+1,   w, w+1,   w,   1, 0],
        [  1, w+1,   0,   1, w+1, 1],
        [w+1, w+1,   w, w+1,   1, 1]
    ])
    m3209 = QuaternaryMatroid(reduced_matrix=A, groundset=['b0','a','y','z','x','c0','b','c','a0'])
    m3209.rename('FN9: ' + repr(m3209))
    return m3209

#An excluded minor for G- and K_2-representable matroids.
#UPF is (Q(a,b), <-1,a,b,a-1,b-1,a-b,a+b,a+b-2,a+b-2ab>).
#(An excluded minor for H_2- and GF(5)-representable matroids.)
def FX9():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [  0, w+1,   0,   w,   1],
        [  1,   w,   1, w+1,   1],
        [w+1, w+1,   w, w+1, w+1],
        [  w,   w, w+1, w+1,   1]
    ])
    m48806 = QuaternaryMatroid(reduced_matrix=A)
    m48806.rename('FX9: ' + repr(m48806))
    return m48806

#An excluded minor for G-representable matroids (and GF(5)-representable matroids.)
#In a DY-equivalence class of 4 matroids.  Has a KP8-minor (delete 8).  UPF is GF(4).
def KR9():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [w+1,   w, w+1,   w, 1],
        [  0,   1, w+1,   w, 1],
        [  w, w+1,   1,   1, 1],
        [w+1, w+1,   w, w+1, 0]
    ])
    kr9 = QuaternaryMatroid(reduced_matrix=A, groundset=[2,4,0,6,1,5,3,7,8])
    kr9.rename('KR9: ' + repr(kr9))
    return kr9

#An excluded minor for G-representable matroids (and GF(5)-representable matroids.)
#Has a TQ8-minor (delete 6) and a KP8-minor (delete 8).  UPF is GF(4).
def KQ9():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [w+1, w, w+1,   1, w+1],
        [  1, 0, w+1, w+1,   1],
        [  0, 1,   w, w+1,   1],
        [  1, 1, w+1,   0, w+1]
    ])
    kq9 = QuaternaryMatroid(reduced_matrix=A, groundset=[5,0,4,3,2,6,8,7,1])
    kq9.rename('KQ9: ' + repr(kq9))
    return kq9

### 10 elements:

#An excluded minor for K_2- and P_4-representable matroids.  Self-dual.
#An excluded minor for H_3- and H_2-representable matroids.  Uniquely GF(5)-representable.
#Although not P_4-representable, it is O-representable, and hence is representable over all fields of size at least four.
def UG10():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [1,   0, 1, w, w+1],
        [1,   1, 1, 1,   1],
        [1,   0, w, 1, w+1],
        [1, w+1, w, 1,   0],
        [1,   1, 1, 0,   0]
    ])
    ug10 = QuaternaryMatroid(reduced_matrix=A)
    ug10.rename('UG10: ' + repr(ug10))
    return ug10

#An excluded minor for K_2-representable matroids.  UPF is P_4.  Self-dual.
def FF10():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [  0,   0, 1, 1,   1],
        [  1,   0, 0, 1,   1],
        [1+w,   1, 0, 0,   1],
        [  1,   1, w, 0,   1],
        [1+w, 1+w, w, w, 1+w]
    ])
    ff10 = QuaternaryMatroid(reduced_matrix=A, groundset=[0,1,2,3,4,5,6,7,8,9])
    ff10.rename('FF10: ' + repr(ff10))
    return ff10

#An excluded minor for K_2-representable matroids.  UPF is G.  Self-dual.
def GP10():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [w+1,   w,   0, 1,   1],
        [  w,   w,   0, 0,   1],
        [  0,   0, w+1, w,   1],
        [  1,   0,   w, 0,   1],
        [  1,   1,   1, 1,   0]
    ])
    gp10 = QuaternaryMatroid(reduced_matrix=A)
    gp10.rename('GP10: ' + repr(gp10))
    return gp10

#An excluded minor for K_2- and G-representable matroids (and H2- and GF(5)-representable matroids).
#UPF is W.  Not self-dual.
def FZ10():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [0, 0,   1,   w,   1],
        [1, 0,   w, w+1, w+1],
        [1, 1,   w,   w,   w],
        [0, 1,   0,   w,   w],
        [1, 1, w+1,   1,   w]
    ])
    fz10 = QuaternaryMatroid(reduced_matrix=A)
    fz10.rename('FZ10: ' + repr(fz10))
    return fz10

#An excluded minor for K_2- and G-representable matroids (and H2- and GF(5)-representable matroids).  Self-dual.
# UPF is (Q(a,b), <-1,a,b,a-1,b-1,a-b,a+b,a+1,ab+b-1,ab-b+1>)
def UQ10():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [  0,   0, 1,   w, 1],
        [  1,   1, 1,   1, 0],
        [  1, w+1, 0,   1, 0],
        [w+1,   w, 0,   0, 1],
        [  1,   1, 1, w+1, 1]
    ])
    uq10 = QuaternaryMatroid(reduced_matrix=A)
    uq10.rename('UQ10: ' + repr(uq10))
    return uq10

#An excluded minor for K_2- and G-representable matroids (and H2- and GF(5)-representable matroids).  UPF is U_1^(2).  Self-dual.
def FP10():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [1, 1,     1, 1, 1],
        [0, 1,     1, 1, 0],
        [w, w, w + 1, 0, 0],
        [0, 1,     1, 0, w],
        [1, w,     1, w, 1]
    ])
    fp10 = QuaternaryMatroid(reduced_matrix=A)
    fp10.rename('FP10: ' + repr(fp10))
    return fp10

#An excluded minor for K_2-representable matroids.  UPF is G.  Self-dual.
#Has TQ8 as a minor (delete 'd' and contract 'c').
def TQ10():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [  1,   w,   0, w+1, 1],
        [w+1, w+1, w+1,   w, 1],
        [w+1,   w,   1,   0, 1],
        [  1,   1,   0, w+1, 0],
        [w+1,   0,   w, w+1, 0]
    ])
    tp10 = QuaternaryMatroid(reduced_matrix=A, groundset=[1,6,8,'c',3,7,'d',2,5,4])
    tp10.rename('TQ10: ' + repr(tp10))
    return tp10

#An excluded minor for P_4-representable matroids.  UPF is G.  Not self-dual.
def FY10():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [1,   1,   1,   0, 0],
        [0,   1, w+1,   1, 0],
        [1,   1,   1, w+1, 1],
        [0,   0,   w,   1, 1],
        [1, w+1,   1,   1, w]
    ])
    fy10 = QuaternaryMatroid(reduced_matrix=A)
    fy10.rename('FY10: ' + repr(fy10))
    return fy10

#An excluded minor for P_4-representable matroids.  UPF is U_1^(2).
#Has a TQ8-minor (e.g. delete 'a' and contract 'e') and a PP9 (and hence P8p) minor (contract 'x')
def PP10():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [w+1,   0, w+1,   0,   w],
        [  w,   w,   1,   w,   1],
        [w+1, w+1,   0, w+1,   1],
        [  1,   0,   1, w+1,   1],
        [  w,   1, w+1,   w, w+1]
    ])
    pp10 = QuaternaryMatroid(reduced_matrix=A, groundset=['z','f','c','g','e','b','a','h','d','x'])
    pp10.rename('PP10: ' + repr(pp10))
    return pp10

#An excluded minor for P_4-representable matroids.  UPF is G.  Self-dual.
def FU10():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [  0,   1,   w,   1,   1],
        [  w, w+1,   1,   1,   w],
        [  1,   1,   0, w+1,   0],
        [  1,   1,   0,   w, w+1],
        [w+1,   w, w+1,   w,   0]
    ])
    fu10 = QuaternaryMatroid(reduced_matrix=A)
    fu10.rename('FU10: ' + repr(fu10))
    return fu10

#An excluded minor for P_4-representable matroids.  UPF is G.  Has a TQ8-minor.
#In a DY*-equivalence class of 13 matroids.
def D10():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [  w, 1,   w,   1, w+1,   w],
        [  w, 0, w+1, w+1,   w,   w],
        [w+1, 0,   0, w+1, w+1, w+1],
        [w+1, 1,   0,   1,   w,   0]
    ])
    d10 = QuaternaryMatroid(reduced_matrix=A)
    d10.rename('D10: ' + repr(d10))
    return d10

#An excluded minor for G-representable matroids (and GF(5)-representable matroids). Not self-dual.
#UPF is GF(4).
def UK10():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [  1,   w, w+1,   w, w+1],
        [  1,   w,   w, w+1,   w],
        [  1, w+1, w+1,   0,   w],
        [  1,   1,   w,   0,   w],
        [w+1,   0,   0,   1,   w]
    ])
    uk10 = QuaternaryMatroid(reduced_matrix=A)
    uk10.rename('UK10: ' + repr(uk10))
    return uk10

#An excluded minor for G-representable matroids (and GF(5)-representable matroids). Not self-dual.
#UPF is GF(4).
def PK10():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [  1,   1, w,   0, w+1],
        [  0,   1, 1,   1,   1],
        [w+1,   w, w, w+1,   w],
        [  0,   1, w, w+1,   1],
        [w+1, w+1, 0,   0,   1]
    ])
    pk10 = QuaternaryMatroid(reduced_matrix=A)
    pk10.rename('PK10: ' + repr(pk10))
    return pk10

#An excluded minor for G-representable matroids (and GF(5)-representable matroids). Not self-dual.
#UPF is GF(4).
def GK10():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [1,   1,   0,   1,   1],
        [1,   0,   0,   w,   w],
        [w, w+1,   1,   1,   0],
        [w, w+1, w+1, w+1, w+1],
        [1,   w,   w, w+1,   1]
    ])
    gk10 = QuaternaryMatroid(reduced_matrix=A)
    gk10.rename('GK10: ' + repr(gk10))
    return gk10

#An excluded minor for G-representable matroids (and GF(5)-representable matroids).  Self-dual.
#UPF is GF(4).
def FT10():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [w, 0,   w, w+1, w+1],
        [0, 1, w+1,   w,   1],
        [w, 1,   0, w+1, w+1],
        [w, 1,   0,   0,   w],
        [0, 1,   w,   0,   1]
    ])
    ft10 = QuaternaryMatroid(reduced_matrix=A)
    ft10.rename('FT10: ' + repr(ft10))
    return ft10

#An excluded minor for G-representable matroids (and GF(5)-representable matroids).  Self-dual.
#UPF is GF(4).
def TK10():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [  1, 1,   0,   0,   w],
        [  0, w,   0, w+1,   1],
        [w+1, 0, w+1,   w, w+1],
        [  w, 0,   1,   0,   w],
        [  0, w,   1,   w,   0]
    ])
    tk10 = QuaternaryMatroid(reduced_matrix=A)
    tk10.rename('TK10: ' + repr(tk10))
    return tk10

#An excluded minor for G-representable matroids (and GF(5)-representable matroids).  Self-dual.
#UPF is GF(4).
def KT10():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [  0,   1,   1,   w,   0],
        [  0,   1,   0, w+1,   1],
        [w+1,   0,   1, w+1, w+1],
        [  w, w+1, w+1,   1,   w],
        [w+1, w+1, w+1,   1,   1]
    ])
    kt10 = QuaternaryMatroid(reduced_matrix=A)
    kt10.rename('KT10: ' + repr(kt10))
    return kt10

#An excluded minor for G-representable matroids (and GF(5)-representable matroids).  Self-dual.
#UPF is GF(4).
def TU10():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [  1, w+1,   1,   0,   w],
        [  w,   0,   1, w+1, w+1],
        [  w,   w, w+1, w+1,   1],
        [w+1,   0, w+1,   1, w+1],
        [w+1, w+1,   1,   w,   1]
    ])
    tu10 = QuaternaryMatroid(reduced_matrix=A)
    tu10.rename('TU10: ' + repr(tu10))
    return tu10

#An excluded minor for G-representable matroids (and GF(5)-representable matroids).  Self-dual.
#UPF is I.
def UT10():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [  w, w+1, 0, w+1,   0],
        [w+1, w+1, 1,   w,   1],
        [  1,   0, 1, w+1, w+1],
        [w+1, w+1, 1,   1,   w],
        [  1, w+1, 1,   w, w+1]
    ])
    ut10 = QuaternaryMatroid(reduced_matrix=A)
    ut10.rename('UT10: ' + repr(ut10))
    return ut10

#An excluded minor for G-representable matroids (and GF(5)-representable matroids).  Self-dual.
#UPF is GF(4).
def FK10():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [  0,   0,   1, 1,   w],
        [  w,   1,   w, 1,   1],
        [  1,   1,   0, 0, w+1],
        [w+1,   0, w+1, 0,   1],
        [w+1, w+1,   1, 1, w+1]
    ])
    fk10 = QuaternaryMatroid(reduced_matrix=A)
    fk10.rename('FK10: ' + repr(fk10))
    return fk10

#An excluded minor for G-representable matroids (and GF(5)-representable matroids).  Self-dual.
#UPF is GF(4).
def KF10():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [w+1, w+1, 1,   w,   1],
        [  0, w+1, w,   1,   0],
        [  0,   w, 0,   1, w+1],
        [w+1,   0, w, w+1,   1],
        [w+1,   1, 0, w+1,   0]
    ])
    kf10 = QuaternaryMatroid(reduced_matrix=A)
    kf10.rename('KF10: ' + repr(kf10))
    return kf10

### 11 elements:

#An excluded minor for P_4-representable matroids.  UPF is PT.
#In a DY*-equivalence class of 6 matroids.  Has a FF10-minor (delete 10).
def FA11():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [  w,   0,   w,   w, 1, 0],
        [  1,   w,   0,   w, w, 1],
        [  0,   w,   1, w+1, 0, w],
        [  0,   w,   0, w+1, 0, 1],
        [w+1, w+1, w+1,   0, w, 0]
    ])
    fa11 = QuaternaryMatroid(reduced_matrix=A, groundset=[1,3,4,2,8,7,9,0,5,10,6])
    fa11.rename('FA11: ' + repr(fa11))
    return fa11

### 12 elements:

#An excluded minor for K_2-representable matroids.  UPF is P_4.  Self-dual.
def FR12():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [0, 1, 1,   1,   0,   1],
        [1, 1, 1,   0,   1,   1],
        [1, 1, 0,   0,   0,   1],
        [1, 0, 0,   0, w+1,   1],
        [0, 1, 0, w+1,   0,   1],
        [1, 1, 1,   1,   1, w+1]
    ])
    fr12 = QuaternaryMatroid(reduced_matrix=A)
    fr12.rename('FR12: ' + repr(fr12))
    return fr12

#An excluded minor for K_2-representable matroids.  UPF is G.  Not self-dual.
def GP12():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [1+w, 1+w,   w, 1+w,   0,   0],
        [  1,   0,   w, 1+w,   0,   0],
        [  1,   0,   1,   0, 1+w, 1+w],
        [  1,   1,   0,   1,   w,   w],
        [  1,   0,   1,   1,   0, 1+w],
        [  1,   1,   1,   1,   1, 1+w]
    ])
    gp12 = QuaternaryMatroid(reduced_matrix=A)
    gp12.rename('GP12: ' + repr(gp12))
    return gp12

#An excluded minor for P_4-representable matroids.  UPF is PT.
#Has a PP9- (contract 4 and 7, delete 6) and FF10-minor (contract 'c' and delete 'd').
def FQ12():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [0, 0,   w,   w,   1, 0],
        [0, 0, w+1, w+1,   1, 1],
        [1, 1,   w,   1,   1, 1],
        [w, 0,   1,   1,   0, 0],
        [w, w, w+1, w+1,   1, 1],
        [0, 1,   1,   w, w+1, 1]
    ])
    fq12 = QuaternaryMatroid(reduced_matrix=A, groundset=[7,4,5,9,2,1,0,6,'d','c',8,3])
    fq12.rename('FQ12: ' + repr(fq12))
    return fq12

#An excluded minor for P_4-representable matroids.  Self-dual.
#UPF is (Q(a,b),<-1,a,b,a-2,a-1,a+1,b-1,ab-a+b,ab-a-b,ab-a-2b>)
#Has a FF10-minor (contract 'c' and delete 'd').
def FF12():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [1,   1, 0, 1,   0, 0],
        [1,   1, w, 1,   0, 1],
        [w, w+1, 1, 0,   1, 1],
        [1,   1, w, 1, w+1, 0],
        [1,   1, 0, 0, w+1, 0],
        [1, w+1, 1, 0, w+1, w]
    ])
    ff12 = QuaternaryMatroid(reduced_matrix=A, groundset=[0,4,'c',3,5,'d',8,9,2,7,1,6])
    ff12.rename('FF12: ' + repr(ff12))
    return ff12

#An excluded minor for K_2- and G-representable matroids (and H2- and GF(5)-representable matroids).  UPF is W.  Not self-dual.
def FZ12():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [  1, w+1, 0, w+1, 0, w+1],
        [w+1, w+1, w,   0, w,   1],
        [  w,   1, 0,   w, 0,   w],
        [w+1,   1, 1, w+1, 1,   w],
        [  w,   w, 1,   0, 0,   0],
        [w+1,   1, 0,   w, 1,   w]
    ])
    fz12 = QuaternaryMatroid(reduced_matrix=A)
    fz12.rename('FZ12: ' + repr(fz12))
    return fz12

#An excluded minor for K_2- and G-representable matroids (and H2- and GF(5)-representable matroids).  UPF is P_pappus.  Self-dual.
def UQ12():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [  1, 0,   0, w+1, 1,   0],
        [w+1, w, w+1,   1, 1,   1],
        [w+1, w, w+1, w+1, w,   1],
        [  1, 0,   0,   w, 1, w+1],
        [  1, 0,   1,   w, 1,   w],
        [  1, 1,   1,   w, 1, w+1]
    ])
    uq12 = QuaternaryMatroid(reduced_matrix=A)
    uq12.rename('UQ12: ' + repr(uq12))
    return uq12

#An excluded minor for K_2- and G-representable matroids (and H2- and GF(5)-representable matroids).  UPF is W.  Self-dual.
def FP12():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [  0, w+1,   1,   0,   1,   w],
        [  0, w+1,   1,   0, w+1,   0],
        [w+1,   1,   w,   0, w+1,   w],
        [  w,   1,   w,   1,   w, w+1],
        [w+1,   0, w+1,   w, w+1,   0],
        [  w,   0, w+1, w+1,   w,   0]
    ])
    fp12 = QuaternaryMatroid(reduced_matrix=A)
    fp12.rename('FP12: ' + repr(fp12))
    return fp12

#An excluded minor for G-representable matroids (and GF(5)-representable matroids).  Rank 5.
#UPF is GF(4).
def FS12():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [1,   1,   1, 1,   1, 0, 1],
        [0,   1, w+1, w, w+1, 1, 1],
        [w, w+1,   0, 0, w+1, 0, 0],
        [0,   0, w+1, 0,   0, 1, w],
        [1,   0,   w, w,   w, 1, 0]
    ])
    fs12 = QuaternaryMatroid(reduced_matrix=A)
    fs12.rename('FS12: ' + repr(fs12))
    return fs12

#An excluded minor for G-representable matroids (and GF(5)-representable matroids).  Self-dual.
#UPF is I.
def UK12():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [0, w+1,   1,   0,   0,   w],
        [w,   1,   w, w+1,   w, w+1],
        [1, w+1, w+1,   0,   0,   0],
        [0,   w,   1,   1, w+1,   1],
        [0,   w,   1,   1,   0, w+1],
        [w,   w, w+1,   w,   w,   1]
    ])
    uk12 = QuaternaryMatroid(reduced_matrix=A)
    uk12.rename('UK12: ' + repr(uk12))
    return uk12

#An excluded minor for G-representable matroids (and GF(5)-representable matroids).  Not self-dual.
#UPF is GF(4).
def UA12():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [0, w+1,   w, w+1, w+1, 0],
        [1,   1,   1, w+1,   w, 1],
        [1, w+1,   0,   1, w+1, 1],
        [0,   0,   0, w+1,   w, 1],
        [0, w+1,   0,   w, w+1, 1],
        [1,   w, w+1, w+1,   w, 1]
    ])
    ua12 = QuaternaryMatroid(reduced_matrix=A)
    ua12.rename('UA12: ' + repr(ua12))
    return ua12

#An excluded minor for G-representable matroids (and GF(5)-representable matroids).  Not self-dual.
#UPF is GF(4).
def AK12():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [  0,  w,   0, 0, w+1,   w],
        [w+1,  w,   0, w, w+1, w+1],
        [  0,  w, w+1, 0,   w,   0],
        [  1,  0,   w, 0,   0, w+1],
        [  0,  1,   0, 1,   1,   0],
        [  1,  0,   1, 1,   0,   1]
    ])
    ak12 = QuaternaryMatroid(reduced_matrix=A)
    ak12.rename('AK12: ' + repr(ak12))
    return ak12

#An excluded minor for G-representable matroids (and GF(5)-representable matroids).  Self-dual.
#UPF is GF(4).
def FK12():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [w+1,   w, w,   0,   w, w+1],
        [  w,   1, 1, w+1, w+1,   w],
        [  1, w+1, w,   0, w+1,   0],
        [  w,   1, 1,   w, w+1, w+1],
        [w+1, w+1, w,   0,   w,   0],
        [  1,   w, w,   w,   1,   1]
    ])
    fk12 = QuaternaryMatroid(reduced_matrix=A)
    fk12.rename('FK12: ' + repr(fk12))
    return fk12

#An excluded minor for G-representable matroids (and GF(5)-representable matroids).  Self-dual.
#UPF is GF(4).
def KB12():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [1, 0,   w, 0, 0,   1],
        [1, w,   0, w, 1, w+1],
        [1, 1, w+1, 0, 1, w+1],
        [w, w,   1, w, 0,   0],
        [1, 1,   1, 0, 0,   0],
        [w, w,   1, w, 1,   1]
    ])
    kb12 = QuaternaryMatroid(reduced_matrix=A)
    kb12.rename('KB12: ' + repr(kb12))
    return kb12

#An excluded minor for G-representable matroids (and GF(5)-representable matroids).  Self-dual.
#UPF is GF(4).
def AF12():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [1, 0,   0,   0,   1,   1],
        [0, 1,   0,   w,   w, w+1],
        [0, 1,   0,   0, w+1, w+1],
        [w, 1,   1,   0,   0,   1],
        [0, 1, w+1,   w,   w,   1],
        [1, 1,   1, w+1, w+1,   w]
    ])
    af12 = QuaternaryMatroid(reduced_matrix=A)
    af12.rename('AF12: ' + repr(af12))
    return af12

#utility function
def complement(groundset, subset):
    return list(set(groundset).difference(subset))

#A matroid with no U(2,4)-detachable pairs (only {ei,fi} pairs are detachable)
def nestOfTwistedCubes():
    gs=["e1","e2","e3","e4","e5","e6","f1","f2","f3","f4","f5","f6"]
    ntc = Matroid(groundset=gs, circuit_closures={3:[
        ["e1", "e2", "f3", "f4"], ["e3", "e4", "f5", "f6"], ["e5", "e6", "f1", "f2"], 
        ["e4", "e5", "f1", "f3"], ["e1", "e3", "f6", "f2"], ["e6", "e2", "f4", "f5"], 
        ["e2", "e5", "f3", "f6"], ["e1", "e4", "f2", "f5"], ["e3", "e6", "f1", "f4"], 
        ["e5", "e1", "f4", "f6"], ["e4", "e6", "f2", "f3"], ["e2", "e3", "f5", "f1"], 
        ["e2", "e4", "f6", "f1"], ["e6", "e1", "f3", "f5"], ["e3", "e5", "f2", "f4"]], 5:[
        complement(gs, ["e1", "e2", "f5", "f6"]), complement(gs, ["e3", "e4", "f1", "f2"]), complement(gs, ["e5", "e6", "f3", "f4"]), 
        complement(gs, ["e4", "e5", "f6", "f2"]), complement(gs, ["e1", "e3", "f4", "f5"]), complement(gs, ["e6", "e2", "f1", "f3"]), 
        complement(gs, ["e2", "e5", "f1", "f4"]), complement(gs, ["e1", "e4", "f3", "f6"]), complement(gs, ["e3", "e6", "f2", "f5"]), 
        complement(gs, ["e5", "e1", "f2", "f3"]), complement(gs, ["e4", "e6", "f5", "f1"]), complement(gs, ["e2", "e3", "f4", "f6"]), 
        complement(gs, ["e2", "e4", "f3", "f5"]), complement(gs, ["e6", "e1", "f2", "f4"]), complement(gs, ["e3", "e5", "f6", "f1"])], 6:[gs]})
    return ntc

### 13 elements:

#An excluded minor for G-representable matroids (and GF(5)-representable matroids).
#UPF is GF(4)
def XY13():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [0,   0,   1,   1,   0, 1,   1],
        [w,   1,   w, w+1,   1, 1, w+1],
        [0,   0, w+1,   1,   1, w,   1],
        [0,   w,   1,   1, w+1, 1,   1],
        [w, w+1,   w, w+1,   1, 1,   w],
        [1,   0,   0,   1,   0, w,   0]
    ])
    xy13 = QuaternaryMatroid(reduced_matrix=A)
    xy13.rename('XY13: ' + repr(xy13))
    return xy13

### 14 elements:

#An excluded minor for dyadic matroids (and GF(5)-representable matroids).
#UPF is GF(3).  4- (but not 5-) connected.  Self-dual.
def N3():
    A = Matrix(GF(3), [
        [2, 0, 0, 2, 1, 1, 2],
        [1, 2, 0, 0, 2, 0, 2],
        [0, 1, 2, 1, 0, 0, 2],
        [0, 2, 2, 0, 0, 0, 2],
        [1, 0, 0, 0, 1, 0, 2],
        [2, 0, 0, 1, 2, 1, 1],
        [1, 1, 1, 2, 2, 2, 0]
    ])
    n3 = TernaryMatroid(reduced_matrix=A)
    n3.rename('N3: ' + repr(n3))
    return n3

#An excluded minor for K_2-representable matroids.  Self-dual.
#Obtained by relaxing the two complementary circuit-hyperplanes of N4.
#Not P_4-representable, but O-representable, and hence representable over all fields of size at least four.
def N3pp():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [0,     1,     0,     1,     0,     w,     w],
        [1,     w,     1,     0,     1,     0,     1],
        [1,     0,     0,     0,     w,     w,     w],
        [1,     0,     0,     0,     1,     0,     1],
        [1,     1,     1,     0,     1,     0,     0],
        [1,     w,     1,     w,     1,     1,     1],
        [1,     1,     0,     1,     0,     0,     0]
    ])
    sq14 = QuaternaryMatroid(reduced_matrix=A)
    sq14.rename('N3=: ' + repr(sq14))
    return sq14

#An excluded minor for K_2-representable matroids.  Has disjoint circuit-hyperplanes.  UPF is W.  Not self-dual.
def UP14():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [1, w, w, 1, w, 0, 0],
        [w, w, w, 1, w, w, 0],
        [w, w, 1, w, 1, 0, 1],
        [1, 1, 0, 1, 0, 0, 0],
        [w, w, 1, w, w, w, 1],
        [0, w, 1, 1, 0, 0, 0],
        [1, 1, 1, 1, 1, 1, 1],
    ])
    up14 =  QuaternaryMatroid(reduced_matrix=A)
    up14.rename('UP14: ' + repr(up14))
    return up14

#An excluded minor for K_2-representable matroids.  Has disjoint circuit-hyperplanes.  UPF is W.  Not self-dual.
def VP14():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [w, 0, 1, 1, w, 1, 1],
        [w, w, 1, 1, w, 1, w],
        [1, 1, 0, 1, 1, w, 0],
        [1, 1, 0, 1, 0, 0, 0],
        [0, 0, 0, w, 0, 1, 1],
        [1, 0, 0, w, 1, 0, 0],
        [1, 1, 1, 1, w, 1, 0]
    ])
    vp14 = QuaternaryMatroid(reduced_matrix=A)
    vp14.rename('VP14: ' + repr(vp14))
    return vp14

#An excluded minor for P_4-representable matroids.  Not self-dual.  UPF is PT
def FV14():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [0,   1,  0,  1,   1,   1,  1],
        [0,   1,  0,  1,   0,   1,  0],
        [1,   0,  0,  1,   1,   1,  1],
        [0, 1-w,  1,  1, 1-w, 1-w,  1],
        [0,   0,  1,  0, 1-w,  -w,  1],
        [1-w, 0, -1, -w,   0,   0, -w],
        [1,   0,  0,  1,   0,   0,  1]
    ])
    fv14 = QuaternaryMatroid(reduced_matrix=A)
    fv14.rename('FV14: ' + repr(fv14))
    return fv14

#An excluded minor for P_4-representable matroids.  Self-dual.  UPF is Orthrus
def OW14():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [0,     1,     1,     w,     1,     0,     w],
        [0, w + 1,     1,     w, w + 1,     0, w + 1],
        [0, w + 1,     1,     w,     0, w + 1, w + 1],
        [1,     1, w + 1, w + 1,     0,     1,     1],
        [1,     0,     0,     w,     0,     0,     w],
        [1,     0,     1,     0,     1,     0,     0],
        [0,     1,     0,     w,     0,     1,     0]
    ])
    ow14 = QuaternaryMatroid(reduced_matrix=A)
    ow14.rename('OW14: ' + repr(ow14))
    return ow14

#An excluded minor for P_4-representable matroids.  Self-dual.  UPF is PT
def FM14():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [1, 0, w, 1, 1, w, 0],
        [0, 1, 0, 0, 1, 1, 1],
        [0, 1, 0, 0, w, w, 1],
        [w, 1, 0, w, 0, 1, 1],
        [0, 1, 1, 0, 0, 0, 1],
        [1, w, w, 0, 0, 0, 0],
        [0, w, 0, 1, 0, w, 0]
    ])
    fm14 = QuaternaryMatroid(reduced_matrix=A)
    fm14.rename('FM14: ' + repr(fm14))
    return fm14

### 15 elements:
#An excluded minor for O-representable matroids.  UPF is PT.
#In a DY*-equivalence class of 6 matroids.  Has an SQ14-minor.
def FA15():
    gf4 = GF(4,'w')
    w = gf4('w')
    A = Matrix(gf4, [
        [1, 1, 1, 1, 1, 1, 1, 1],
        [1, 0, 1, 1, 1, 1, 0, 1],
        [0, 0, 1, 0, 0, 1, 0, 1],
        [0, 0, w, w, 0, 1, 1, 1],
        [1, 1, 0, 0, 0, 0, w, w],
        [w, 0, 1, 1, 1, 1, 0, 0],
        [w, w, 1, 1, 0, 0, 0, 0]
    ])
    fa15 = QuaternaryMatroid(reduced_matrix=A)
    fa15.rename('FA15: ' + repr(fa15))
    return fa15

### 16 elements:

#An excluded minor for dyadic matroids (and GF(5)-representable matroids).
#UPF is GF(3).  4- (but not 5-) connected.  Self-dual.
def N4():
    A = Matrix(GF(3), [
        [2, 0, 2, 1, 2, 1, 0, 0],
        [2, 1, 0, 2, 0, 2, 2, 0],
        [2, 0, 2, 0, 0, 1, 0, 0],
        [2, 0, 2, 2, 2, 2, 2, 2],
        [0, 1, 1, 1, 1, 1, 1, 1],
        [2, 1, 0, 0, 0, 2, 0, 0],
        [2, 0, 0, 2, 2, 2, 2, 0],
        [1, 0, 1, 2, 1, 2, 1, 1]
    ])
    n4 = TernaryMatroid(reduced_matrix=A)
    n4.rename('N4: ' + repr(n4))
    return n4