cayley_graphs_boolean_dimension_2.ipynb

Kernel: SageMath 9.7

Cayley graphs of binary bent functions of dimension 2.

Import the required modules.

from os import path
from boolean_cayley_graphs.bent_function import BentFunction
from boolean_cayley_graphs.bent_function_cayley_graph_classification import (
     BentFunctionCayleyGraphClassification)
from boolean_cayley_graphs.boolean_function_extended_translate_classification import (
     BooleanFunctionExtendedTranslateClassification)

Import controls.

import boolean_cayley_graphs.cayley_graph_controls as controls

Turn on verbose output.

controls.timing = True
controls.verbose = True

Connect to the database that contains the classifications of bent functions in 2 dimensions.

load(path.join("..", "sage-code", "boolean_dimension_cayley_graph_classifications.py"))

Set c to be the list of classifications for dimension 2, starting from 1. c[0] is None.

dim = 2

sobj_dir = path.join("..", "sobj")

c = save_boolean_dimension_cayley_graph_classifications(dim, dir=sobj_dir)

Function 1 :
2025-05-28 09:12:39.542556 0 0
2025-05-28 09:12:39.554080 1 2
2025-05-28 09:12:39.570954 2 2
2025-05-28 09:12:39.577357 3 2
2025-05-28 09:12:39.584686
Algebraic normal form of Boolean function: x0*x1
Function is bent.


SDP design incidence structure t-design parameters: (True, (1, 4, 1, 1))

Classification of Cayley graphs and classification of Cayley graphs of duals are the same:

There are 2 extended Cayley classes in the extended translation class.

Display the length of c, the list of classifications.

len(c)

2

Verify that c[0] is None.

print(c[0])

None

Print the algebraic normal form of the bent function corresponding to c[1].

n = 1

c[n].algebraic_normal_form

x0*x1

Produce a report on the classification c[1].

c[n].report(report_on_graph_details=True)

Algebraic normal form of Boolean function: x0*x1
Function is bent.


SDP design incidence structure t-design parameters: (True, (1, 4, 1, 1))

Classification of Cayley graphs and classification of Cayley graphs of duals are the same:

There are 2 extended Cayley classes in the extended translation class.

For each extended Cayley class in the extended translation class:
Clique polynomial, strongly regular parameters, rank, and order of a representative graph; and
linear code and generator matrix for a representative bent function:

EC class 0 :
Algebraic normal form of representative: x0*x1
Clique polynomial: 2*t^2 + 4*t + 1
Strongly regular parameters: (4, 1, 0, 0)
Rank: 4 Order: 8

Linear code from representative:
[1, 1] linear code over GF(2)
Generator matrix:
[1]
Linear code is projective.
Weight distribution: {0: 1, 1: 1}

EC class 1 :
Algebraic normal form of representative: x0*x1 + x0 + x1
Clique polynomial: t^4 + 4*t^3 + 6*t^2 + 4*t + 1
Strongly regular parameters: False
Rank: 4 Order: 24

Linear code from representative:
[3, 2] linear code over GF(2)
Generator matrix:
[1 0 1]
[0 1 1]
Linear code is projective.
Weight distribution: {0: 1, 2: 3}

Produce a matrix plot of the weight_class_matrix.

matrix_plot(c[n].weight_class_matrix, cmap='gist_stern', colorbar=True)

Produce a matrix plot of bent_cayley_graph_index_matrix, the matrix of indices of extended Cayley classes within the extended translation class.

matrix_plot(c[n].bent_cayley_graph_index_matrix, cmap='gist_stern', colorbar=True)

%%time

bf = BentFunction(c[n].algebraic_normal_form)
bf_etc = BooleanFunctionExtendedTranslateClassification.from_function(bf)

2025-05-28 09:12:47.962490 0 0
2025-05-28 09:12:47.967929 1 4
2025-05-28 09:12:47.980397 2 4
2025-05-28 09:12:47.985798 3 4
2025-05-28 09:12:47.990917
CPU times: user 24.9 ms, sys: 10 ms, total: 34.9 ms
Wall time: 37.9 ms

name_n = "p" + str(dim) + "_" + str(n)
bf_etc.save_mangled(name_n, dir=sobj_dir)

print("Number of bent functions in the extended translation class is", 
      len(bf_etc.boolean_function_list))
print("Number of general linear equivalence classes in the extended translation class is", 
      len(bf_etc.general_linear_class_list))

Number of bent functions in the extended translation class is 4
Number of general linear equivalence classes in the extended translation class is 2

matrix_plot(bf_etc.general_linear_class_index_matrix, cmap='gist_stern', colorbar=True)

matrix_plot(bf_etc.boolean_function_index_matrix, cmap='gist_stern', colorbar=True)