Kernel: SageMath 9.7
Cayley graphs of binary bent functions of dimension 4.
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 4 dimensions.
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Set c
to be the list of classifications for dimension 4, starting from 1. c[0]
is None
.
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Function 1 :
2023-02-19 09:54:45.443634 0 0
2023-02-19 09:54:46.016205 1 2
2023-02-19 09:54:46.062804 2 2
2023-02-19 09:54:46.108978 3 2
2023-02-19 09:54:46.161200 4 2
2023-02-19 09:54:46.207955 5 2
2023-02-19 09:54:46.263597 6 2
2023-02-19 09:54:46.313619 7 2
2023-02-19 09:54:46.364364 8 2
2023-02-19 09:54:46.414189 9 2
2023-02-19 09:54:46.473091 10 2
2023-02-19 09:54:46.523071 11 2
2023-02-19 09:54:46.578541 12 2
2023-02-19 09:54:46.633258 13 2
2023-02-19 09:54:46.697818 14 2
2023-02-19 09:54:46.745101 15 2
2023-02-19 09:54:46.798626
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.
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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 + x2*x3
Produce a report on the classification c[1]
.
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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
.
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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 09:54:48.413747 0 0
2023-02-19 09:54:48.515655 1 16
2023-02-19 09:54:48.584020 2 16
2023-02-19 09:54:48.642784 3 16
2023-02-19 09:54:48.712505 4 16
2023-02-19 09:54:48.790483 5 16
2023-02-19 09:54:48.856198 6 16
2023-02-19 09:54:48.920643 7 16
2023-02-19 09:54:48.989903 8 16
2023-02-19 09:54:49.064421 9 16
2023-02-19 09:54:49.130729 10 16
2023-02-19 09:54:49.205797 11 16
2023-02-19 09:54:49.278383 12 16
2023-02-19 09:54:49.337300 13 16
2023-02-19 09:54:49.397667 14 16
2023-02-19 09:54:49.458843 15 16
2023-02-19 09:54:49.520858
CPU times: user 1.02 s, sys: 12 ms, total: 1.03 s
Wall time: 1.12 s
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Number of bent functions in the extended translation class is 16
Number of general linear equivalence classes in the extended translation class is 2
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