Classification of bent functions by their Cayley graphs

The bent_function_cayley_graph_classification module defines:

  • the BentFunctionCayleyGraphClassification class; which represents the classification of the Cayley graphs within the extended translation class of a bent function; and

  • the BentFunctionCayleyGraphClassPart class, which represents part of a Cayley graph classification.

AUTHORS:

  • Paul Leopardi (2016-08-02): initial version

EXAMPLES:

The classification of the bent function defined by the polynomial x2 + x1*x2.

sage: from boolean_cayley_graphs.bent_function import BentFunction
sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
sage: R2.<x1,x2> = BooleanPolynomialRing(2)
sage: p = x2+x1*x2
sage: f = BentFunction(p)
sage: c = BentFunctionCGC.from_function(f)
sage: dict(sorted(c.__dict__.items()))
{'algebraic_normal_form': x0*x1 + x1,
 'bent_cayley_graph_index_matrix': [0 0 1 0]
 [1 0 0 0]
 [0 0 0 1]
 [0 1 0 0],
 'cayley_graph_class_list': ['CK', 'C~'],
 'dual_cayley_graph_index_matrix': [0 0 1 0]
 [1 0 0 0]
 [0 0 0 1]
 [0 1 0 0],
 'weight_class_matrix': [0 0 1 0]
 [1 0 0 0]
 [0 0 0 1]
 [0 1 0 0]}

REFERENCES:

The extended translation equivalence class and the extended Cayley equivalence class of a bent function are defined by Leopardi [Leo2017].

class boolean_cayley_graphs.bent_function_cayley_graph_classification.BentFunctionCayleyGraphClassPart(*args, **kwargs)[source]

Bases: sage.structure.sage_object.SageObject, boolean_cayley_graphs.saveable.Saveable

Partial classification of the Cayley graphs within the extended translation equivalence class of a bent function.

EXAMPLES:

sage: from boolean_cayley_graphs.bent_function import BentFunction
sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassPart as BentFunctionCGCP
sage: R2.<x1,x2> = BooleanPolynomialRing(2)
sage: p = x1+x2+x1*x2
sage: f = BentFunction(p)
sage: c1 = BentFunctionCGCP.from_function(f, c_stop=1)
sage: print(c1)
BentFunctionCayleyGraphClassPart.from_function(BentFunction(x0*x1 + x0 + x1, c_start=0, c_stop=1))
sage: latex(c1)
\text{\texttt{BentFunctionCayleyGraphClassPart.from{\char`\_}function(BentFunction(x0*x1{ }+{ }x0{ }+{ }x1,{ }c{\char`\_}start=0,{ }c{\char`\_}stop=1))}}
classmethod from_function(bentf, list_dual_graphs=True, c_start=0, c_stop=None, limited_memory=False, algorithm='sage')[source]

Constructor from the BentFunction bentf.

INPUT:

  • bentf – an object of class BentFunction.

  • list_dual_graphs – boolean (default: True). A flag indicating whether to list dual graphs.

  • c_start – integer (default: 0). The smallest value of c to use for extended translates.

  • c_stop – integer (default: None). One more than largest value of c to use for extended translates. None means use all remaining values.

  • limited_memory – boolean (default: False). A flag indicating whether the classification might be too large to fit into memory.

  • algorithm – string (default: default_algorithm). The algorithm used for canonical labelling.

OUTPUT:

An object of class BentFunctionCayleyGraphClassPart, initialized as follows.

  • algebraic_normal_form is set to bentf.algebraic_normal_form(),

  • cayley_graph_class_list is set to a list of graph6_string stings corresponding to the complete set of non-isomorphic Cayley graphs of the bent functions within the extended translation equivalence class of bentf (and their duals, if list_dual_graphs is True),

  • bent_cayley_graph_index_matrix is set to a matrix of indices into cayley_graph_class_list corresponding to these bent functions,

  • dual_cayley_graph_index_matrix is set to None if list_dual_graphs is False, otherwise it is set to a matrix of indices into cayley_graph_class_list corresponding to the duals of these bent functions, and

  • weight_class_matrix is set to the 0-1 matrix of weight classes corresponding to bent_cayley_graph_index_matrix,

  • c_start is set to smallest value of c used for extended translates.

Each entry bent_cayley_graph_index_matrix[c-c_start,b] corresponds to the Cayley graph of the bent function \(x \mapsto \mathtt{bentf}(x+b) + \langle c, x \rangle + \mathtt{bentf}(b)\). This enumerates all of the extended translates of bentf having c from c_start to but not including c_stop.

EXAMPLES:

A partial classification of the bent function defined by the polynomial \(x_1 + x_2 + x_1 x_2\).

sage: from boolean_cayley_graphs.bent_function import BentFunction
sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassPart as BentFunctionCGCPart
sage: R2.<x1,x2> = BooleanPolynomialRing(2)
sage: p = x1+x2+x1*x2
sage: f = BentFunction(p)
sage: c1 = BentFunctionCGCPart.from_function(f,c_start=2,c_stop=4)
sage: dict(sorted(c1.__dict__.items()))
{'algebraic_normal_form': x0*x1 + x0 + x1,
'bent_cayley_graph_index_matrix': [0 1 0 0]
[0 0 0 1],
'c_start': 2,
'cayley_graph_class_list': ['CK', 'C~'],
'dual_cayley_graph_index_matrix': [0 1 0 0]
[0 0 0 1],
'weight_class_matrix': [0 1 0 0]
[0 0 0 1]}

A partial classification of the bent function defined by the polynomial \(x_1 + x_2 + x_1 x_2\), but with list_dual_graphs=False.

sage: from boolean_cayley_graphs.bent_function import BentFunction
sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
sage: R2.<x1,x2> = BooleanPolynomialRing(2)
sage: p = x1+x2+x1*x2
sage: f = BentFunction(p)
sage: c2 = BentFunctionCGCPart.from_function(f,list_dual_graphs=False,c_start=0,c_stop=2)
sage: dict(sorted(c2.__dict__.items()))
{'algebraic_normal_form': x0*x1 + x0 + x1,
'bent_cayley_graph_index_matrix': [0 1 1 1]
[1 1 0 1],
'c_start': 0,
'cayley_graph_class_list': ['C~', 'CK'],
'dual_cayley_graph_index_matrix': None,
'weight_class_matrix': [1 0 0 0]
[0 0 1 0]}
class boolean_cayley_graphs.bent_function_cayley_graph_classification.BentFunctionCayleyGraphClassification(*args, **kwargs)[source]

Bases: boolean_cayley_graphs.bent_function_cayley_graph_classification.BentFunctionCayleyGraphClassPart

Classification of the Cayley graphs within the extended translation equivalence class of a bent function.

EXAMPLES:

sage: from boolean_cayley_graphs.bent_function import BentFunction
sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
sage: R2.<x1,x2> = BooleanPolynomialRing(2)
sage: p = x1+x2+x1*x2
sage: f = BentFunction(p)
sage: c1 = BentFunctionCGC.from_function(f)
sage: print(c1)
BentFunctionCayleyGraphClassification.from_function(BentFunction(x0*x1 + x0 + x1))
sage: latex(c1)
\text{\texttt{BentFunctionCayleyGraphClassification.from{\char`\_}function(BentFunction(x0*x1{ }+{ }x0{ }+{ }x1))}}
bent_function_csv_suffix = '_bent_function.csv'
cg_class_list_csv_suffix = '_cg_class_list.csv'
classmethod cg_class_list_from_csv(file_name)[source]

Read a Cayley graph class list from a csv file.

The csv file is assumed to be created by the method save_cg_class_list_as_csv().

INPUT:

  • file_name – the name of the csv file.

OUTPUT:

A list of Cayley graphs in graph6_string format.

EXAMPLES:

sage: from boolean_cayley_graphs.bent_function import BentFunction
sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
sage: import os
sage: bf2 = BentFunction([1,0,1,1])
sage: c2 = BentFunctionCGC.from_function(bf2)
sage: csv_name = tmp_filename(ext=".csv")
sage: c2.save_cg_class_list_as_csv(csv_name)
sage: cgcl_saved = BentFunctionCGC.cg_class_list_from_csv(csv_name)
sage: print(cgcl_saved == c2.cayley_graph_class_list)
True
sage: os.remove(csv_name)
first_matrix_index_list()[source]

Obtain a representative bent function corresponding to each extended Cayley class.

INPUT:

  • self – the current object.

OUTPUT:

A list of tuples (i_n,j_n), each of which is the first index into the matrix self.bent_cayley_graph_index_matrix that contains the entry n. The first index is determined by argwhere.

EXAMPLES:

The result for the bent function defined by the polynomial \(x_1 + x_2 + x_1 x_2\).

sage: from boolean_cayley_graphs.bent_function import BentFunction
sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
sage: R2.<x1,x2> = BooleanPolynomialRing(2)
sage: p = x1+x2+x1*x2
sage: f = BentFunction(p)
sage: c = BentFunctionCGC.from_function(f)
sage: c.first_matrix_index_list()
[(0, 0), (0, 1)]
classmethod from_csv(file_name_prefix)[source]

Constructor from three csv files.

INPUT:

  • file_name_prefix – string: the common prefix to use for file names.

OUTPUT:

None.

EXAMPLES:

sage: from boolean_cayley_graphs.bent_function import BentFunction
sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
sage: import os
sage: bf2 = BentFunction([1,1,0,1])
sage: c2 = BentFunctionCGC.from_function(bf2)
sage: prefix = tmp_filename()
sage: c2.save_as_csv(prefix)
sage: c2_saved = BentFunctionCGC.from_csv(prefix)
sage: print(c2 == c2_saved)
True
sage: bent_function_csv_name = prefix + BentFunctionCGC.bent_function_csv_suffix
sage: os.remove(bent_function_csv_name)
sage: cg_class_list_csv_name = prefix + BentFunctionCGC.cg_class_list_csv_suffix
sage: os.remove(cg_class_list_csv_name)
sage: matrices_csv_name = prefix + BentFunctionCGC.matrices_csv_suffix
sage: os.remove(matrices_csv_name)
classmethod from_function(bentf, list_dual_graphs=True, limited_memory=False)[source]

Constructor from the BentFunction bentf.

INPUT:

  • bentf – an object of class BentFunction.

  • list_dual_graphs – boolean. a flag indicating whether to list dual graphs.

  • limited_memory – boolean. A flag indicating whether the classification might be too large to fit into memory. Default is False.

OUTPUT:

An object of class BentFunctionCayleyGraphClassification, initialized as follows.

  • algebraic_normal_form is set to bentf.algebraic_normal_form(),

  • cayley_graph_class_list is set to a list of graph6_string strings corresponding to the complete set of non-isomorphic Cayley graphs of the bent functions within the extended translation equivalence class of bentf (and their duals, if list_dual_graphs is True),

  • bent_cayley_graph_index_matrix is set to a matrix of indices into cayley_graph_class_list corresponding to these bent functions,

  • dual_cayley_graph_index_matrix is set to None if list_dual_graphs is False, otherwise it is set to a matrix of indices into cayley_graph_class_list corresponding to the duals of these bent functions, and

  • weight_class_matrix is set to the 0-1 matrix of weight classes corresponding to bent_cayley_graph_index_matrix.

Each entry bent_cayley_graph_index_matrix[c,b] corresponds to the Cayley graph of the bent function \(x \mapsto \mathtt{bentf}(x+b) + \langle c, x \rangle + \mathtt{bentf}(b)\). This enumerates all of the extended translates of bentf.

EXAMPLES:

The classification of the bent function defined by the polynomial \(x_1 + x_2 + x_1 x_2\).

sage: from boolean_cayley_graphs.bent_function import BentFunction
sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
sage: R2.<x1,x2> = BooleanPolynomialRing(2)
sage: p = x1+x2+x1*x2
sage: f = BentFunction(p)
sage: c3 = BentFunctionCGC.from_function(f)
sage: dict(sorted(c3.__dict__.items()))
{'algebraic_normal_form': x0*x1 + x0 + x1,
'bent_cayley_graph_index_matrix': [0 1 1 1]
[1 1 0 1]
[1 0 1 1]
[1 1 1 0],
'cayley_graph_class_list': ['C~', 'CK'],
'dual_cayley_graph_index_matrix': [0 1 1 1]
[1 1 0 1]
[1 0 1 1]
[1 1 1 0],
'weight_class_matrix': [1 0 0 0]
[0 0 1 0]
[0 1 0 0]
[0 0 0 1]}

TESTS:

The classification of the bent function defined by the polynomial \(x_1 + x_2 + x_1 x_2\), but with list_dual_graphs=False.

sage: from boolean_cayley_graphs.bent_function import BentFunction
sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
sage: R2.<x1,x2> = BooleanPolynomialRing(2)
sage: p = x1+x2+x1*x2
sage: f = BentFunction(p)
sage: c4 = BentFunctionCGC.from_function(f,list_dual_graphs=False)
sage: dict(sorted(c4.__dict__.items()))
{'algebraic_normal_form': x0*x1 + x0 + x1,
'bent_cayley_graph_index_matrix': [0 1 1 1]
[1 1 0 1]
[1 0 1 1]
[1 1 1 0],
'cayley_graph_class_list': ['C~', 'CK'],
'dual_cayley_graph_index_matrix': None,
'weight_class_matrix': [1 0 0 0]
[0 0 1 0]
[0 1 0 0]
[0 0 0 1]}
classmethod from_parts(prefix_basename, dir=None, limited_memory=False)[source]

Constructor from saved class parts.

INPUTS:

  • prefix_basename – string. The prefix to use with mangled_name() to obtain the file names of the saved class parts.

  • dir – string, optional. The directory where the parts are located. Default is None, meaning the current directory.

  • limited_memory – boolean, default is False. A flag indicating whether the classification might be too large to fit into memory.

OUTPUT:

An object of class BentFunctionCayleyGraphClassification, constructed from the saved class parts.

EXAMPLES:

A classification of the bent function defined by the polynomial \(x_1 + x_2 + x_1 x_2\).

sage: from boolean_cayley_graphs.bent_function import BentFunction
sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassPart as BentFunctionCGCPart
sage: import os.path
sage: R2.<x1,x2> = BooleanPolynomialRing(2)
sage: p = x1+x2+x1*x2
sage: f = BentFunction(p)
sage: prefix = tmp_filename()
sage: prefix_dirname = os.path.dirname(prefix)
sage: prefix_basename = os.path.basename(prefix)
sage: for row in range(4):
....:     c = BentFunctionCGCPart.from_function(f, c_start=row,c_stop=row+1)
....:     part_prefix = prefix_basename + "_" + str(row)
....:     c.save_mangled(
....:         part_prefix,
....:         dir=prefix_dirname)
sage: cl1 = BentFunctionCGC.from_parts(
....:    prefix_basename,
....:    dir=prefix_dirname)
sage: cl1.report(report_on_matrix_details=True)
Algebraic normal form of Boolean function: x0*x1 + x0 + x1
Function is bent.
<BLANKLINE>
Weight class matrix:
[1 0 0 0]
[0 0 1 0]
[0 1 0 0]
[0 0 0 1]
<BLANKLINE>
SDP design incidence structure t-design parameters: (True, (1, 4, 1, 1))
<BLANKLINE>
Classification of Cayley graphs and classification of Cayley graphs of duals are the same:
<BLANKLINE>
There are 2 extended Cayley classes in the extended translation class.
<BLANKLINE>
Matrix of indices of Cayley graphs:
[0 1 1 1]
[1 1 0 1]
[1 0 1 1]
[1 1 1 0]
sage: for row in range(4):
....:     part_prefix = prefix_basename + "_" + str(row)
....:     BentFunctionCGCPart.remove_mangled(
....:         part_prefix,
....:         dir=prefix_dirname)
matrices_csv_suffix = '_matrices.csv'
classmethod matrices_from_csv(dim, file_name)[source]

Read three matrices from a csv file.

The csv file is assumed to be created by the method save_matrices_as_csv().

INPUT:

  • dim – integer: the dimension of the bent function.

  • file_name – the name of the csv file.

OUTPUT:

A tuple of matrices,

EXAMPLES:

sage: from boolean_cayley_graphs.bent_function import BentFunction
sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
sage: import os
sage: bf2 = BentFunction([1,1,0,1])
sage: dim = bf2.nvariables()
sage: c2 = BentFunctionCGC.from_function(bf2)
sage: csv_name = tmp_filename(ext=".csv")
sage: c2.save_matrices_as_csv(csv_name)
sage: (ci_matrix,di_matrix,wc_matrix) = BentFunctionCGC.matrices_from_csv(dim, csv_name)
sage: print(c2.bent_cayley_graph_index_matrix == ci_matrix)
True
sage: print(c2.dual_cayley_graph_index_matrix == di_matrix)
True
sage: print(c2.weight_class_matrix == wc_matrix)
True
sage: os.remove(csv_name)

TESTS:

Test the case where list_dual_graphs==False and dual_cayley_graph_index_matrix is None.

sage: from boolean_cayley_graphs.bent_function import BentFunction
sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
sage: import os
sage: bf = BentFunction([1,1,0,1])
sage: dim = bf.nvariables()
sage: c = BentFunctionCGC.from_function(bf, list_dual_graphs=False)
sage: csv_name = tmp_filename(ext=".csv")
sage: c.save_matrices_as_csv(csv_name)
sage: (ci_matrix,di_matrix,wc_matrix) = BentFunctionCGC.matrices_from_csv(dim, csv_name)
sage: print(c.bent_cayley_graph_index_matrix == ci_matrix)
True
sage: print(c.dual_cayley_graph_index_matrix == di_matrix)
True
sage: print(c.weight_class_matrix == wc_matrix)
True
sage: os.remove(csv_name)
print_latex_table_of_cayley_classes(width=40, rows_per_table=6)[source]

Print a table of Cayley classes in LaTeX format.

For a given classification, print, in LaTeX format, the table of selected properties of the Cayley classes of that classification.

INPUT:

  • self – the current object.

  • width – integer (default: 40): the table width.

  • rows_per_table – integer (default: 6). The number of rows to include before starting a new table.

OUTPUT:

(To standard output.) A table in LaTeX format.

EXAMPLES:

Print the table of Cayley classes for the classification of the bent function defined by the polynomial \(x_0 + x_0 x_1 + x_2 x_3\).

sage: from boolean_cayley_graphs.bent_function import BentFunction
sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
sage: R4.<x0,x1,x2,x3> = BooleanPolynomialRing(4)
sage: p = x0+x0*x1+x2*x3
sage: f = BentFunction(p)
sage: c = BentFunctionCGC.from_function(f)
sage: c.print_latex_table_of_cayley_classes()
\small{}
\begin{align*}
\def\arraystretch{1.2}
\begin{array}{|cccl|}
\hline
\text{Class} &
\text{Parameters} &
\text{2-rank} &
\text{Clique polynomial}
\\
\hline
0 &
(16, 6, 2, 2) &
6 &
\begin{array}{l}
8t^{4} + 32t^{3} + 48t^{2} + 16t + 1
\end{array}
\\
1 &
(16, 10, 6, 6) &
6 &
\begin{array}{l}
16t^{5} + 120t^{4} + 160t^{3}
\,+
\\
80t^{2} + 16t + 1
\end{array}
\\
\hline
\end{array}
\end{align*}
print_latex_table_of_tonchev_graphs(width=40)[source]

Print a table comparing Cayley graphs with graphs from Tonchev’s codes.

Tonchev’s codes are binary projective two-weight codes as published by Tonchev [Ton1996], [Ton2007].

INPUT:

  • self – the current object.

  • width – integer (default: 40): the table width.

  • algorithm – string (default: default_algorithm). Algorithm used for canonical labelling.

OUTPUT:

(To standard output.) A table in LaTeX format.

Note

The comparison displayed in this table really only makes sense for bent functions in 6 dimensions.

EXAMPLES:

The classification for the bent function defined by the polynomial \(x_0 x_1 + x_2 x_3 + x_4 x_5\).

sage: from boolean_cayley_graphs.bent_function import BentFunction
sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
sage: R6.<x0,x1,x2,x3,x4,x5> = BooleanPolynomialRing(6)
sage: p = x0*x1 + x2*x3 + x4*x5
sage: f = BentFunction(p)
sage: c = BentFunctionCGC.from_function(f) # long time (60 seconds)
sage: c.print_latex_table_of_tonchev_graphs() # long time (depends on line above)
\begin{align*}
\def\arraystretch{1.2}
\begin{array}{|ccl|}
\hline
\text{Class} &
\text{Parameters} &
\text{Reference}
\\
\hline
0 & [35,6,16] & \text{Table 1.156 1 (complement)}
\\
0 & [35,6,16] & \text{Table 1.156 2 (complement)}
\\
1 & [27,6,12] & \text{Table 1.155 1 }
\\
\hline
\end{array}
\end{align*}

REFERENCES:

report(report_on_matrix_details=False, report_on_graph_details=False)[source]

Print a report on the classification.

The report includes attributes and various computed quantities.

INPUT:

  • self – the current object.

  • report_on_matrix_details – Boolean (default: False).

    If True, print each matrix.

  • report_on_graph_details – Boolean (default: False).

    If True, produce a detailed report for each Cayley graph.

OUTPUT:

(To standard output) A report on the following attributes of self:

  • algebraic_normal_form

  • cayley_graph_class_list

  • If report_on_matrix_details is True:

    • bent_cayley_graph_index_matrix

    • dual_cayley_graph_index_matrix (only if this is not None and is different from bent_cayley_graph_index_matrix)

    • weight_class_matrix

  • If report_on_graph_details is True: details of each graph in cayley_graph_class_list.

EXAMPLES:

Report on the classification of the bent function defined by the polynomial \(x_0 + x_0 x_1 + x_2 x_3\).

sage: from boolean_cayley_graphs.bent_function import BentFunction
sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
sage: R4.<x0,x1,x2,x3> = BooleanPolynomialRing(4)
sage: p = x0+x0*x1+x2*x3
sage: f = BentFunction(p)
sage: c = BentFunctionCGC.from_function(f)
sage: c.report(report_on_matrix_details=True, report_on_graph_details=True)
Algebraic normal form of Boolean function: x0*x1 + x0 + x2*x3
Function is bent.
<BLANKLINE>
Weight class matrix:
[0 1 0 0 0 1 0 0 0 1 0 0 1 0 1 1]
[0 0 0 1 0 0 0 1 0 0 0 1 1 1 1 0]
[1 0 0 0 1 0 0 0 1 0 0 0 0 1 1 1]
[0 0 1 0 0 0 1 0 0 0 1 0 1 1 0 1]
[0 1 0 0 1 0 1 1 0 1 0 0 0 1 0 0]
[0 0 0 1 1 1 1 0 0 0 0 1 0 0 0 1]
[1 0 0 0 0 1 1 1 1 0 0 0 1 0 0 0]
[0 0 1 0 1 1 0 1 0 0 1 0 0 0 1 0]
[0 1 0 0 0 1 0 0 1 0 1 1 0 1 0 0]
[0 0 0 1 0 0 0 1 1 1 1 0 0 0 0 1]
[1 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0]
[0 0 1 0 0 0 1 0 1 1 0 1 0 0 1 0]
[1 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0]
[1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 1]
[0 1 1 1 1 0 0 0 1 0 0 0 1 0 0 0]
[1 1 0 1 0 0 1 0 0 0 1 0 0 0 1 0]
<BLANKLINE>
SDP design incidence structure t-design parameters: (True, (2, 16, 6, 2))
<BLANKLINE>
Classification of Cayley graphs and classification of Cayley graphs of duals are the same:
<BLANKLINE>
There are 2 extended Cayley classes in the extended translation class.
<BLANKLINE>
Matrix of indices of Cayley graphs:
[0 1 0 0 0 1 0 0 0 1 0 0 1 0 1 1]
[0 0 0 1 0 0 0 1 0 0 0 1 1 1 1 0]
[1 0 0 0 1 0 0 0 1 0 0 0 0 1 1 1]
[0 0 1 0 0 0 1 0 0 0 1 0 1 1 0 1]
[0 1 0 0 1 0 1 1 0 1 0 0 0 1 0 0]
[0 0 0 1 1 1 1 0 0 0 0 1 0 0 0 1]
[1 0 0 0 0 1 1 1 1 0 0 0 1 0 0 0]
[0 0 1 0 1 1 0 1 0 0 1 0 0 0 1 0]
[0 1 0 0 0 1 0 0 1 0 1 1 0 1 0 0]
[0 0 0 1 0 0 0 1 1 1 1 0 0 0 0 1]
[1 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0]
[0 0 1 0 0 0 1 0 1 1 0 1 0 0 1 0]
[1 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0]
[1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 1]
[0 1 1 1 1 0 0 0 1 0 0 0 1 0 0 0]
[1 1 0 1 0 0 1 0 0 0 1 0 0 0 1 0]
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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:
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EC class 0 :
Algebraic normal form of representative: x0*x1 + x0 + 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
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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}
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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
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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}

REFERENCES:

save_as_csv(file_name_prefix)[source]

Save the classification as three csv files with a common prefix.

INPUT:

  • self – the current object.

  • file_name_prefix – string: the common prefix to use for file names.

OUTPUT:

None.

EXAMPLES:

sage: from boolean_cayley_graphs.bent_function import BentFunction
sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
sage: import os
sage: bf2 = BentFunction([0,0,0,1])
sage: c2 = BentFunctionCGC.from_function(bf2)
sage: prefix = tmp_filename()
sage: c2.save_as_csv(prefix)
sage: c2_saved = BentFunctionCGC.from_csv(prefix)
sage: print(c2 == c2_saved)
True
sage: bent_function_csv_name = prefix + BentFunctionCGC.bent_function_csv_suffix
sage: os.remove(bent_function_csv_name)
sage: cg_class_list_csv_name = prefix + BentFunctionCGC.cg_class_list_csv_suffix
sage: os.remove(cg_class_list_csv_name)
sage: matrices_csv_name = prefix + BentFunctionCGC.matrices_csv_suffix
sage: os.remove(matrices_csv_name)
save_cg_class_list_as_csv(file_name)[source]

Save the Cayley graph class list to a csv file.

INPUT:

  • file_name – the name of the csv file.

OUTPUT:

None.

EXAMPLES:

sage: from boolean_cayley_graphs.bent_function import BentFunction
sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
sage: import os
sage: bf2 = BentFunction([1,0,1,1])
sage: c2 = BentFunctionCGC.from_function(bf2)
sage: csv_name = tmp_filename(ext=".csv")
sage: c2.save_cg_class_list_as_csv(csv_name)
sage: cgcl_saved = BentFunctionCGC.cg_class_list_from_csv(csv_name)
sage: print(cgcl_saved == c2.cayley_graph_class_list)
True
sage: os.remove(csv_name)
save_matrices_as_csv(file_name)[source]

Save the matrices bent_cayley_graph_index_matrix, dual_cayley_graph_index_matrix and weight_class_matrix to a csv file.

INPUT:

  • file_name – the name of the csv file.

OUTPUT:

None.

EXAMPLES:

sage: from boolean_cayley_graphs.bent_function import BentFunction
sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
sage: import os
sage: bf2 = BentFunction([0,1,1,1])
sage: dim = bf2.nvariables()
sage: c2 = BentFunctionCGC.from_function(bf2)
sage: csv_name = tmp_filename(ext=".csv")
sage: c2.save_matrices_as_csv(csv_name)
sage: (ci_matrix,di_matrix,wc_matrix) = BentFunctionCGC.matrices_from_csv(dim, csv_name)
sage: print(c2.bent_cayley_graph_index_matrix == ci_matrix)
True
sage: print(c2.dual_cayley_graph_index_matrix == di_matrix)
True
sage: print(c2.weight_class_matrix == wc_matrix)
True
sage: os.remove(csv_name)

TESTS:

Test the case where list_dual_graphs=False and dual_cayley_graph_index_matrix is None.

sage: from boolean_cayley_graphs.bent_function import BentFunction
sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
sage: import os
sage: bf = BentFunction([1,0,1,1])
sage: dim = bf.nvariables()
sage: c = BentFunctionCGC.from_function(bf, list_dual_graphs=False)
sage: csv_name = tmp_filename(ext=".csv")
sage: c.save_matrices_as_csv(csv_name)
sage: (ci_matrix,di_matrix,wc_matrix) = BentFunctionCGC.matrices_from_csv(dim, csv_name)
sage: print(c.bent_cayley_graph_index_matrix == ci_matrix)
True
sage: print(c.dual_cayley_graph_index_matrix == di_matrix)
True
sage: print(c.weight_class_matrix == wc_matrix)
True
sage: os.remove(csv_name)
save_matrix_plots(figure_name, cmap='gist_stern')[source]

Plot the matrix attributes to figure files.

Use matrix_plot to plot the matrix attributes bent_cayley_graph_index_matrix, dual_cayley_graph_index_matrix, and weight_class_matrix to corresponding figure files.

INPUT:

  • self – the current object.

  • figure_name – string. The prefix to use in the file names for the figures.

  • cmap – string (default: 'gist_stern'). The colormap to use with matrixplot.

OUTPUT:

(To figure files: figure_name + "_bent_cayley_graph_index_matrix.png", figure_name + "_dual_cayley_graph_index_matrix.png", figure_name + "_weight_class_matrix.png") Plots of the corresponding matrix attributes.

EXAMPLES:

sage: from boolean_cayley_graphs.bent_function import BentFunction
sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
sage: import glob
sage: import os
sage: bf = BentFunction([1,1,0,1])
sage: dim = bf.nvariables()
sage: c = BentFunctionCGC.from_function(bf)
sage: figure_name = tmp_filename()
sage: c.save_matrix_plots(figure_name)
sage: figure_list = glob.glob(figure_name+"*.png")
sage: for figure in figure_list:
....:     print(
....:         "_bent_cayley_graph_index_matrix.png" in figure or
....:         "_dual_cayley_graph_index_matrix.png" in figure or
....:         "_weight_class_matrix.png" in figure)
....:     print(os.path.isfile(figure))
....:     os.remove(figure)
True
True
True
True
True
True