r"""

Return a companion matrix in `GL(3, K)` whose multiplicative order is `q^3 - 1`.

EXAMPLES::

sage: m = random_q3_minus_one_matrix(GF(3))

sage: m.multiplicative_order() == 3**3 - 1

True

sage: m = random_q3_minus_one_matrix(GF(4,'a'))

sage: m.multiplicative_order() == 4**3 - 1

True

sage: m = random_q3_minus_one_matrix(GF(5))

sage: m.multiplicative_order() == 5**3 - 1

True

sage: m = random_q3_minus_one_matrix(GF(9,'a'))

sage: m.multiplicative_order() == 9**3 - 1

True

"""

q = K.cardinality()

M = MatrixSpace(K, 3)

if q.is_prime():

from sage.rings.finite_rings.conway_polynomials import conway_polynomial

try:

a,b,c,_ = conway_polynomial(q, 3)

except RuntimeError: # the polynomial is not in the database

pass

else:

return M([0,0,-a,1,0,-b,0,1,-c])

while True:

a = K._random_nonzero_element()

b = K.random_element()

c = K.random_element()

m = M([0,0,-a,1,0,-b,0,1,-c])

if m.multiplicative_order() == q**3 - 1:

r"""

Return the generator of matrix that have Hughes property.

INPUT:

- ``q`` (integer) - the dimension such that generated matrix are in GL(3,GF(q))

EXAMPLES::

sage: G=find_hughes_matrix(3)

sage: G.next()

[0 2 0]

[0 0 2]

[1 2 0]

sage: M = G.next()

sage: M

[0 2 0]

[0 0 2]

[2 2 0]

sage: M** (3**2 + 3 + 1)

[2 0 0]

[0 2 0]

[0 0 2]

"""

M = GL(3,GF(q))

for m in M:

o = gap.Order(m._gap_())

if o >= q**2 + q + 1:

if is_hughes_matrix(m,q,o):

r"""

Check whether the matrix has the Hughes property.

INPUT:

- ``m`` - matrix in GL(3,GF(q))

- ``q`` (integer)

- ``o`` (integer) - m order

EXAMPLES::

sage: G=find_hughes_matrix(3)

sage: M=G.next()

sage: o=gap.Order(M._gap_())

sage: is_hughes_matrix(M,3,o)

True

sage: M** (3**2 + 3 + 1)

[1 0 0]

[0 1 0]

[0 0 1]

sage: M=GL(3,GF(3))[10]

sage: o=gap.Order(M._gap_())

sage: is_hughes_matrix(M,3,o)

False

sage: M ** (3**2 + 3 + 1)

[0 2 1]

[2 1 2]

[2 0 1]

"""

p= m.matrix()**(q**2 + q + 1)

if not p[0,0]==p[1,1]==p[2,2]!=0 or not p[0,1]==p[0,2]==p[1,0]==p[1,2]==p[2,0]==p[2,1]==0:

return False

for i in divisors(int(o)):

if i < q**2 + q + 1:

p = (m**i).matrix()

if p[0,0]==p[1,1]==p[2,2]!=0 and p[0,1]==p[0,2]==p[1,0]==p[1,2]==p[2,0]==p[2,1]==0:

return False

r"""

Return the normalized form of point (x,y,z).

For all integer k non-zero, (x,y,z)k refers to the same point.

For the normalized form, the last non-zero coordinate must be 1.

INPUT:

- ``p`` - point with the coordinates (x,y,z) (a list, a vector, a tuple...)

- ``K```- a finite field (coordinates x,y,z are elements of K)

- ``q`` - cardinality of K

OUTPUT:

List of the coordinates from the normalized form of p

EXAMPLE::

sage: K=FiniteField(9,'x')

sage: p=(K('x'),K('x+1'),K('x'))

sage: normalise(p,K,9)

[1, x, 1]

sage: q=vector((K('x'),K('x'),K('x')))

sage: normalise(q,K,9)

[1, 1, 1]

sage: s=(K('2*x+2'), K(0), K(0))

sage: normalise(s,K,9)

[1, 0, 0]

sage: t=[K('2*x'),K(1),K(0)]

sage: normalise(t,K,9)

[2*x, 1, 0]

"""

if type(p) != list :

p = list(p)

for i in range(3):

if p[2-i] == 1:

break

elif p[2-i] != 0:

k=~p[2-i]

if k.is_square():

for j in range(3-i):

p[j] *= k

for j in range(3-i,3):

p[j] = K(0)

break

else:

for j in range(3-i):

p[j]= p[j]**q * k**q

for j in range(3-i,3):

p[j] = K(0)

break

r"""

Return Hughes projective plane of order ``n2``.

INPUT:

- ``n2`` -- an integer which must be an odd square

EXAMPLES::

sage: HughesPlane(9)

Incidence structure with 91 points and 91 blocks

sage: HughesPlane(9).is_projective_plane()

True

sage: HughesPlane(5)

Traceback (most recent call last):

...

EmptySetError: No Hughes plane of non-square order exists.

sage: HughesPlane(16)

Traceback (most recent call last):

...

EmptySetError: No Hughes plane of even order exists.

"""

if not n2.is_square():

raise EmptySetError("No Hughes plane of non-square order exists.")

if n2%2 == 0:

raise EmptySetError("No Hughes plane of even order exists.")

n = n2.sqrt()

A = find_hughes_matrix(n).next()

K = FiniteField(n2, 'x')

F = FiniteField(n, 'y')

v = K.list()

# Construct the points (x,y,z) of the projective plane, (x,y,z)=(xk,yk,zk)

points=[[x,y,K(1)] for x in v for y in v]+[[x,K(1),K(0)] for x in v]+[[K(1),K(0),K(0)]]

relabel={tuple(p):i for i,p in enumerate(points)}

blcks = []

# Find the first line satisfying x+ay+z=0

for a in v:

if a not in F or a == 1:

l=[]

l.append(vector((-a,K(1),K(0))))

for x in v:

if ((~a)*(-x-K(1))).is_square():

l.append(vector((x,(~a)*(-x-K(1)),K(1))))

else:

l.append(vector((x,(~a) **n * (-x-K(1)),K(1))))

# We can now deduce the other lines from these ones

blcks.append(l)

for i in range(n2 + n):

l = [A*j for j in l]

blcks.append(l)

for b in blcks:

for p in range(len(b)):

b[p]=relabel[tuple(normalize(b[p],K,n))]