Ticket #19221: hyperoval.py

def hyperoval(q):
    r"""
    Return an hyperoval in finite Desarguesian projective planes when q=2^n
    """
    assert q%2==0 and is_prime_power(q)

    q2 = q*q
    K = FiniteField(q, 'x')
    zero = K.zero()
    one  = K.one()
    K_to_int = {x:i for i,x in enumerate(K)}

    # affine plane are the points: K_to_int[x] + q*K_to_int[y]
    # the line at infinity       : q2 + K_to_int[x]
    # the point at infinty       : q2 + q
    
    # build the curve x**2 = yz
    oval = [K_to_int[x] + q*K_to_int[x**2] for x in K_to_int]
    oval.append(q2 + K_to_int[zero] + q*K_to_int[one])

    # add the nucleus
    oval.append(K_to_int[zero])

    return oval

def is_arc(P, a):
    # build the dual
    a = set(a)
    for b in P.blocks():
        if len(a.intersection(b)) > 2:
            return False
    return True


def test():
    for n in range(1,7):
        q = 2**n
        O = hyperoval(q)
        P = designs.DesarguesianProjectivePlaneDesign(q)
        assert is_arc(P,O)