cayley_graphs_boolean_dimension_4.ipynb

Kernel: SageMath 9.7

Cayley graphs of binary bent functions of dimension 4.

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 4 dimensions.

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

Set c to be the list of classifications for dimension 4, starting from 1. c[0] is None.

dim = 4

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

c = save_boolean_dimension_cayley_graph_classifications(dim, dir=sobj_dir)

Function 1 :
2025-05-27 13:54:38.342539 0 0
2025-05-27 13:54:40.551342 1 2
2025-05-27 13:54:40.617041 2 2
2025-05-27 13:54:40.664908 3 2
2025-05-27 13:54:40.714792 4 2
2025-05-27 13:54:40.763340 5 2
2025-05-27 13:54:40.811893 6 2
2025-05-27 13:54:40.859522 7 2
2025-05-27 13:54:40.911734 8 2
2025-05-27 13:54:40.960745 9 2
2025-05-27 13:54:41.008865 10 2
2025-05-27 13:54:41.060062 11 2
2025-05-27 13:54:41.111102 12 2
2025-05-27 13:54:41.162483 13 2
2025-05-27 13:54:41.214008 14 2
2025-05-27 13:54:41.283936 15 2
2025-05-27 13:54:41.380685
Algebraic normal form of Boolean function: x0*x1 + x2*x3
Function is bent.


SDP design incidence structure t-design parameters: (True, (2, 16, 6, 2))

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 + x2*x3

Produce a report on the classification c[1].

c[n].report(report_on_graph_details=True)

Algebraic normal form of Boolean function: x0*x1 + x2*x3
Function is bent.


SDP design incidence structure t-design parameters: (True, (2, 16, 6, 2))

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 + x2*x3
Clique polynomial: 8*t^4 + 32*t^3 + 48*t^2 + 16*t + 1
Strongly regular parameters: (16, 6, 2, 2)
Rank: 6 Order: 1152

Linear code from representative:
[6, 4] linear code over GF(2)
Generator matrix:
[1 0 0 0 0 1]
[0 1 0 1 0 0]
[0 0 1 1 0 0]
[0 0 0 0 1 1]
Linear code is projective.
Weight distribution: {0: 1, 2: 6, 4: 9}

EC class 1 :
Algebraic normal form of representative: x0*x1 + x0 + x1 + x2*x3
Clique polynomial: 16*t^5 + 120*t^4 + 160*t^3 + 80*t^2 + 16*t + 1
Strongly regular parameters: (16, 10, 6, 6)
Rank: 6 Order: 1920

Linear code from representative:
[10, 4] linear code over GF(2)
Generator matrix:
[1 0 1 0 1 0 0 1 0 0]
[0 1 1 0 1 1 0 1 1 0]
[0 0 0 1 1 1 0 0 0 1]
[0 0 0 0 0 0 1 1 1 1]
Linear code is projective.
Weight distribution: {0: 1, 4: 5, 6: 10}

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[1].algebraic_normal_form)
bf_etc = BooleanFunctionExtendedTranslateClassification.from_function(bf)

2025-05-27 13:54:45.241238 0 0
2025-05-27 13:54:45.455595 1 16
2025-05-27 13:54:45.520806 2 16
2025-05-27 13:54:45.592180 3 16
2025-05-27 13:54:45.683511 4 16
2025-05-27 13:54:45.759822 5 16
2025-05-27 13:54:45.834626 6 16
2025-05-27 13:54:45.905379 7 16
2025-05-27 13:54:45.998103 8 16
2025-05-27 13:54:46.076366 9 16
2025-05-27 13:54:46.164385 10 16
2025-05-27 13:54:46.247318 11 16
2025-05-27 13:54:46.322478 12 16
2025-05-27 13:54:46.417283 13 16
2025-05-27 13:54:46.496576 14 16
2025-05-27 13:54:46.569768 15 16
2025-05-27 13:54:46.660503
CPU times: user 1.27 s, sys: 24.1 ms, total: 1.3 s
Wall time: 1.43 s

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 16
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)