r"""
Boolean linear codes
====================
The ``boolean_linear_code`` module defines the functions:
* ``boolean_linear_code_graph``;
which returns the Boolean linear code corresponding to a Boolean function,
* ``linear_code_from_code_gens``;
which return the Boolean linear code corresponding to a list of generators; and
* ``print_latex_code_parameters``,
which prints the standard parameters of a linear code.
AUTHORS:
- Paul Leopardi (2016-10-28): initial version
"""
from sage.coding.linear_code import LinearCode
from sage.matrix.constructor import matrix
from sage.rings.finite_rings.finite_field_constructor import FiniteField as GF
from boolean_cayley_graphs.integer_bits import inner
def boolean_linear_code(dim, f):
r"""
Return the Boolean linear code corresponding to a Boolean function.
INPUT:
- ``dim`` -- positive integer. The assumed dimension of function ``f``.
- ``f`` -- a Python function that takes a positive integer and returns 0 or 1.
This is assumed to represent a Boolean function on :math:`\mathbb{F}_2^{dim}`
via lexicographical ordering.
OUTPUT:
An object of class ``LinearCode``, representing the Boolean linear code
corresponding to the Boolean function represented by ``f``.
EXAMPLES:
::
sage: from sage.crypto.boolean_function import BooleanFunction
sage: bf = BooleanFunction([0,1,0,0,0,1,0,0,1,0,0,0,0,0,1,1])
sage: dim = bf.nvariables()
sage: from boolean_cayley_graphs.boolean_linear_code import boolean_linear_code
sage: bc = boolean_linear_code(dim, bf)
sage: bc.characteristic_polynomial()
-2/3*x + 2
sage: bc.generator_matrix().echelon_form()
[1 0 0 0 1]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 1]
REFERENCES:
.. Carlet [Car2010]_.
.. Calderbank and Kantor [CalK1986]_.
.. Ding [Din2015]_ Corollary 10.
"""
v = 2 ** dim
support = [
y
for y in range(v)
if f(y) == 1]
M = matrix(GF(2), [[
inner(2 ** k, y)
for y in support]
for k in range(dim)])
return LinearCode(M)
def linear_code_from_code_gens(gens):
r"""
Return the Boolean linear code corresponding to a list of generators.
INPUT:
- ``gens`` -- list. A list of strings of 0,1 characters.
This is assumed to represent the generators of a linear code.
OUTPUT:
An object of class ``LinearCode`` representing the Boolean linear code
corresponding to the generators represented by ``gens``.
EXAMPLE:
::
sage: from boolean_cayley_graphs.boolean_linear_code import linear_code_from_code_gens
sage: gens = (
....: "10001",
....: "01000",
....: "00100",
....: "00011")
sage: c = linear_code_from_code_gens(gens)
sage: c.basis()
[
(1, 0, 0, 0, 1),
(0, 1, 0, 0, 0),
(0, 0, 1, 0, 0),
(0, 0, 0, 1, 1)
]
"""
M = matrix(GF(2), [list(s) for s in gens])
return LinearCode(M)
def print_latex_code_parameters(c):
r"""
Print the standard parameters of a linear code.
INPUT:
- ``c`` -- ``LinearCode``.
OUTPUT:
A string representing the standard parameters of the linear code ``c``.
EXAMPLE:
::
sage: from boolean_cayley_graphs.boolean_linear_code import linear_code_from_code_gens
sage: from boolean_cayley_graphs.boolean_linear_code import print_latex_code_parameters
sage: gens = (
....: "10001",
....: "01000",
....: "00100",
....: "00011")
sage: c = linear_code_from_code_gens(gens)
sage: print_latex_code_parameters(c)
[5,4,1]
"""
print((
"[" + str(c.length()) +
"," + str(c.dimension()) +
"," + str(c.minimum_distance()) + "]"), end=' ')
