Python ProjectiveSpace.point_set方法代碼示例

# 需要導入模塊: from sage.schemes.projective.projective_space import ProjectiveSpace [as 別名]
# 或者: from sage.schemes.projective.projective_space.ProjectiveSpace import point_set [as 別名]
    def parity_check_matrix(self):
        r"""
        Return a parity check matrix of ``self``.

        The construction of the parity check matrix in case ``self``
        is not a binary code is not really well documented.
        Regarding the choice of projective geometry, one might check:

        - the note over section 2.3 in [Rot2006]_, pages 47-48
        - the dedicated paragraph in [HP2003]_, page 30

        EXAMPLES::

            sage: C = codes.HammingCode(GF(3), 3)
            sage: C.parity_check_matrix()
            [1 0 1 1 0 1 0 1 1 1 0 1 1]
            [0 1 1 2 0 0 1 1 2 0 1 1 2]
            [0 0 0 0 1 1 1 1 1 2 2 2 2]

        """
        n = self.length()
        F = self.base_field()
        m = n - self.dimension()
        MS = MatrixSpace(F, n, m)
        X = ProjectiveSpace(m - 1, F)
        PFn = [list(p) for p in X.point_set(F).points()]

        H = MS(PFn).transpose()
        H = H[::-1, :]
        H.set_immutable()
        return H

# 需要導入模塊: from sage.schemes.projective.projective_space import ProjectiveSpace [as 別名]
# 或者: from sage.schemes.projective.projective_space.ProjectiveSpace import point_set [as 別名]
def HammingCode(r,F):
    r"""
    Implements the Hamming codes.

    The `r^{th}` Hamming code over `F=GF(q)` is an
    `[n,k,d]` code with length `n=(q^r-1)/(q-1)`,
    dimension `k=(q^r-1)/(q-1) - r` and minimum distance
    `d=3`. The parity check matrix of a Hamming code has rows
    consisting of all nonzero vectors of length r in its columns,
    modulo a scalar factor so no parallel columns arise. A Hamming code
    is a single error-correcting code.

    INPUT:


    -  ``r`` - an integer 2

    -  ``F`` - a finite field.


    OUTPUT: Returns the r-th q-ary Hamming code.

    EXAMPLES::

        sage: codes.HammingCode(3,GF(2))
        Linear code of length 7, dimension 4 over Finite Field of size 2
        sage: C = codes.HammingCode(3,GF(3)); C
        Linear code of length 13, dimension 10 over Finite Field of size 3
        sage: C.minimum_distance()
        3
        sage: C.minimum_distance(algorithm='gap') # long time, check d=3
        3
        sage: C = codes.HammingCode(3,GF(4,'a')); C
        Linear code of length 21, dimension 18 over Finite Field in a of size 2^2

    While the ``codes`` object now gathers all code constructors,
    ``HammingCode`` is still available in the global namespace::

        sage: HammingCode(3,GF(2))
        doctest:...: DeprecationWarning: This method soon will not be available in that way anymore. To use it, you can now call it by typing codes.HammingCode
        See http://trac.sagemath.org/15445 for details.
        Linear code of length 7, dimension 4 over Finite Field of size 2

    """
    q = F.order()
    n =  (q**r-1)/(q-1)
    k = n-r
    MS = MatrixSpace(F,n,r)
    X = ProjectiveSpace(r-1,F)
    PFn = [list(p) for p in X.point_set(F).points(F)]
    H = MS(PFn).transpose()
    Cd = LinearCode(H)
    # Hamming code always has distance 3, so we provide the distance.
    return LinearCode(Cd.dual_code().gen_mat(), d=3)

# 需要導入模塊: from sage.schemes.projective.projective_space import ProjectiveSpace [as 別名]
# 或者: from sage.schemes.projective.projective_space.ProjectiveSpace import point_set [as 別名]
def HammingCode(r,F):
    r"""
    Implements the Hamming codes.

    The `r^{th}` Hamming code over `F=GF(q)` is an
    `[n,k,d]` code with length `n=(q^r-1)/(q-1)`,
    dimension `k=(q^r-1)/(q-1) - r` and minimum distance
    `d=3`. The parity check matrix of a Hamming code has rows
    consisting of all nonzero vectors of length r in its columns,
    modulo a scalar factor so no parallel columns arise. A Hamming code
    is a single error-correcting code.

    INPUT:


    -  ``r`` - an integer 2

    -  ``F`` - a finite field.


    OUTPUT: Returns the r-th q-ary Hamming code.

    EXAMPLES::

        sage: codes.HammingCode(3,GF(2))
        Linear code of length 7, dimension 4 over Finite Field of size 2
        sage: C = codes.HammingCode(3,GF(3)); C
        Linear code of length 13, dimension 10 over Finite Field of size 3
        sage: C.minimum_distance()
        3
        sage: C.minimum_distance(algorithm='gap') # long time, check d=3
        3
        sage: C = codes.HammingCode(3,GF(4,'a')); C
        Linear code of length 21, dimension 18 over Finite Field in a of size 2^2
    """
    q = F.order()
    n =  (q**r-1)/(q-1)
    k = n-r
    MS = MatrixSpace(F,n,r)
    X = ProjectiveSpace(r-1,F)
    PFn = [list(p) for p in X.point_set(F).points(F)]
    H = MS(PFn).transpose()
    Cd = LinearCode(H)
    # Hamming code always has distance 3, so we provide the distance.
    return LinearCode(Cd.dual_code().generator_matrix(), d=3)