Kernel: SageMath 9.7
Cayley graphs of binary bent functions of dimension 2.
Import the required modules.
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Import controls.
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Turn on verbose output.
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Connect to the database that contains the classifications of bent functions in 2 dimensions.
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Set c
to be the list of classifications for dimension 2, starting from 1. c[0]
is None
.
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Function 1 :
2023-02-19 09:53:31.047422 0 0
2023-02-19 09:53:32.269575 1 2
2023-02-19 09:53:32.275485 2 2
2023-02-19 09:53:32.284021 3 2
2023-02-19 09:53:32.292853
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.
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2
Verify that c[0]
is None
.
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None
Print the algebraic normal form of the bent function corresponding to c[1]
.
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x0*x1
Produce a report on the classification c[1]
.
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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
.
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Produce a matrix plot of bent_cayley_graph_index_matrix
, the matrix of indices of extended Cayley classes within the extended translation class.
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2023-02-19 09:53:39.182923 0 0
2023-02-19 09:53:39.189692 1 4
2023-02-19 09:53:39.194997 2 4
2023-02-19 09:53:39.199583 3 4
2023-02-19 09:53:39.204330
CPU times: user 22.3 ms, sys: 8.74 ms, total: 31 ms
Wall time: 28.6 ms
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Number of bent functions in the extended translation class is 4
Number of general linear equivalence classes in the extended translation class is 2
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