Python FiniteField.list方法代码示例

# 需要导入模块: from sage.rings.finite_rings.finite_field_constructor import FiniteField [as 别名]
# 或者: from sage.rings.finite_rings.finite_field_constructor.FiniteField import list [as 别名]

#.........这里部分代码省略.........
      returning it. As this is expected to be useless (but we are cautious
      guys), you may want to disable it whenever you want speed. Set to
      ``True`` by default.

    EXAMPLES::

        sage: H = designs.HughesPlane(9)
        sage: H
        (91,10,1)-Balanced Incomplete Block Design

    We prove in the following computations that the Desarguesian plane ``H`` is
    not Desarguesian. Let us consider the two triangles `(0,1,10)` and `(57, 70,
    59)`. We show that the intersection points `D_{0,1} \cap D_{57,70}`,
    `D_{1,10} \cap D_{70,59}` and `D_{10,0} \cap D_{59,57}` are on the same line
    while `D_{0,70}`, `D_{1,59}` and `D_{10,57}` are not concurrent::

        sage: blocks = H.blocks()
        sage: line = lambda p,q: next(b for b in blocks if p in b and q in b)

        sage: b_0_1 = line(0, 1)
        sage: b_1_10 = line(1, 10)
        sage: b_10_0 = line(10, 0)
        sage: b_57_70 = line(57, 70)
        sage: b_70_59 = line(70, 59)
        sage: b_59_57 = line(59, 57)

        sage: set(b_0_1).intersection(b_57_70)
        {2}
        sage: set(b_1_10).intersection(b_70_59)
        {73}
        sage: set(b_10_0).intersection(b_59_57)
        {60}

        sage: line(2, 73) == line(73, 60)
        True

        sage: b_0_57 = line(0, 57)
        sage: b_1_70 = line(1, 70)
        sage: b_10_59 = line(10, 59)

        sage: p = set(b_0_57).intersection(b_1_70)
        sage: q = set(b_1_70).intersection(b_10_59)
        sage: p == q
        False

    TESTS:

    Some wrong input::

        sage: designs.HughesPlane(5)
        Traceback (most recent call last):
        ...
        EmptySetError: No Hughes plane of non-square order exists.

        sage: designs.HughesPlane(16)
        Traceback (most recent call last):
        ...
        EmptySetError: No Hughes plane of even order exists.

    Check that it works for non-prime `q`::

        sage: designs.HughesPlane(3**4)    # not tested - 10 secs
        (6643,82,1)-Balanced Incomplete Block Design
    """
    if not q2.is_square():
        raise EmptySetError("No Hughes plane of non-square order exists.")
    if q2%2 == 0:
        raise EmptySetError("No Hughes plane of even order exists.")
    q = q2.sqrt()
    K = FiniteField(q2, prefix='x')
    F = FiniteField(q, prefix='y')
    A = q3_minus_one_matrix(F)
    A = A.change_ring(K)
    m = K.list()
    V = VectorSpace(K, 3)
    zero = K.zero()
    one = K.one()
    points = [(x, y, one) for x in m for y in m] + \
             [(x, one, zero) for x in m] + \
             [(one, zero, zero)]
    relabel = {tuple(p):i for i,p in enumerate(points)}
    blcks = []
    for a in m:
        if a not in F or a == 1:
            # build L(a)
            aa = ~a
            l = []
            l.append(V((-a, one, zero)))
            for x in m:
                y = - aa * (x+one)
                if not y.is_square():
                    y *= aa**(q-1)
                l.append(V((x, y, one)))
            # compute the orbit of L(a)
            blcks.append([relabel[normalize_hughes_plane_point(p,q)] for p in l])
            for i in range(q2 + q):
                l = [A*j for j in l]
                blcks.append([relabel[normalize_hughes_plane_point(p,q)] for p in l])
    from .bibd import BalancedIncompleteBlockDesign
    return BalancedIncompleteBlockDesign(q2**2+q2+1, blcks, check=check)