code: def _kernel(binary_matrix):
      r"""Computes the kernel of a binary matrix on the binary finite field :math:`\mathbb{Z}_2`.
  
      Args:
          binary_matrix (array[int]): binary matrix representation of the Hamiltonian
      Returns:
          array[int]: nullspace of the `binary_matrix` where each row corresponds to a
          basis vector in the nullspace
  
      **Example**
  
      >>> binary_matrix = np.array([[1, 0, 0, 0, 0, 1, 0, 0],
      ...                          [0, 0, 1, 1, 1, 1, 1, 1],
      ...                          [0, 0, 0, 1, 1, 0, 0, 1]])
      >>> _kernel(binary_matrix)
      array([[0, 1, 0, 0, 0, 0, 0, 0],
             [0, 0, 1, 1, 1, 0, 0, 0],
             [1, 0, 1, 0, 0, 1, 0, 0],
             [0, 0, 1, 0, 0, 0, 1, 0],
             [0, 0, 1, 1, 0, 0, 0, 1]])
      """
      # Get the columns with and without pivots
      pivots = (binary_matrix.T != 0).argmax(axis=0)
      nonpivots = np.setdiff1d(range(len(binary_matrix[0])), pivots)
  
      # Initialize the nullspace
      null_vector = np.zeros(
          (binary_matrix.shape[1], len(nonpivots)), dtype=int
      )
      null_vector[nonpivots, np.arange(len(nonpivots))] = 1
  
      # Fill up the nullspace vectors from the binary matrix
      null_vector_indices = np.ix_(pivots, np.arange(len(nonpivots)))
      binary_vector_indices = np.ix_(np.arange(len(pivots)), nonpivots)
      null_vector[null_vector_indices] = (
          -binary_matrix[binary_vector_indices] % 2
      )
  
      nullspace = null_vector.T
      return nullspace
  
imports: import functools
  import itertools
  import numpy
  import scipy
  import pennylane
  from pennylane.operation import active_new_opmath
  from pennylane.pauli import PauliSentence, PauliWord, pauli_sentence, simplify
  from pennylane.pauli.utils import _binary_matrix_from_pws
  from pennylane.wires import Wires
label_desc: Use image node_id 2 for calling a global function with example usage: _kernel(binary_matrix) and returns: nullspace