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

boolean_cayley_graphs.boolean_linear_code.boolean_linear_code(dim, f)[source]

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 \(\mathbb{F}_2^{dim}\) 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:

boolean_cayley_graphs.boolean_linear_code.linear_code_from_code_gens(gens)[source]

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)
]
boolean_cayley_graphs.boolean_linear_code.print_latex_code_parameters(c)[source]

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]