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P. Kerntopf et al.
The above 4 representatives of RevFunCFLVs have the following cycle
structures:
< 0 >< 1 >< 2 >< 4 >< 6 >< 8 >< 10 >< 12 >
< 3, 11 >< 5, 7 >< 9, 13 >< 14, 15 >
< 0 >< 8 >< 1, 15 >< 2, 3, 4, 5 >< 6, 9, 7, 14 >< 10, 13, 12, 11 >
< 0 >< 5 >< 4, 10, 6, 13 >< 1, 15, 8, 12, 11, 9, 3, 14, 7, 2 >
< 0 >< 7 >< 8 >< 15 >< 1, 13, 4, 14, 2, 11 >< 3, 10, 6, 12, 5, 9 >,
respectively.
It is possible to extrapolate the information collected by us about 3- and
4-variable RevFunCFLVs in many ways. First, let us note that there are no
simple similarities between cycle structures for 3-variable and 4-variable functions.
However, it can be noticed that the cycle structure of the first of the 4-variable
representatives is the simplest one, consisting of just 4 transpositions (besides 1element cycles):
< 3, 11 >< 5, 7 >< 9, 13 >< 14, 15 >
Let us look closer at these transpositions expressing their elements as binary
strings and change the order of transpositions in the following way:
15
7
13 11
1111 0111 1101 1011
1110 0101 1001 0011
14
5
9
3
In each of the pairs of binary strings forming a transposition the strings differ
exactly in one bit. Moreover, we can observe the following property:
– In the first transposition the value of the first position from right is changing.
– In the second transposition the value of the second position from right is
changing.
– In the third transposition the value of the third position from right is changing.
– In the fourth transposition the value of the fourth position from right is changing.
It is easy to extrapolate these properties and below we show that it leads to the
desired infinite sequence of RevFunCFLVs.
P. Kerntopf et al.
The above 4 representatives of RevFunCFLVs have the following cycle
structures:
< 0 >< 1 >< 2 >< 4 >< 6 >< 8 >< 10 >< 12 >
< 3, 11 >< 5, 7 >< 9, 13 >< 14, 15 >
< 0 >< 8 >< 1, 15 >< 2, 3, 4, 5 >< 6, 9, 7, 14 >< 10, 13, 12, 11 >
< 0 >< 5 >< 4, 10, 6, 13 >< 1, 15, 8, 12, 11, 9, 3, 14, 7, 2 >
< 0 >< 7 >< 8 >< 15 >< 1, 13, 4, 14, 2, 11 >< 3, 10, 6, 12, 5, 9 >,
respectively.
It is possible to extrapolate the information collected by us about 3- and
4-variable RevFunCFLVs in many ways. First, let us note that there are no
simple similarities between cycle structures for 3-variable and 4-variable functions.
However, it can be noticed that the cycle structure of the first of the 4-variable
representatives is the simplest one, consisting of just 4 transpositions (besides 1element cycles):
< 3, 11 >< 5, 7 >< 9, 13 >< 14, 15 >
Let us look closer at these transpositions expressing their elements as binary
strings and change the order of transpositions in the following way:
15
7
13 11
1111 0111 1101 1011
1110 0101 1001 0011
14
5
9
3
In each of the pairs of binary strings forming a transposition the strings differ
exactly in one bit. Moreover, we can observe the following property:
– In the first transposition the value of the first position from right is changing.
– In the second transposition the value of the second position from right is
changing.
– In the third transposition the value of the third position from right is changing.
– In the fourth transposition the value of the fourth position from right is changing.
It is easy to extrapolate these properties and below we show that it leads to the
desired infinite sequence of RevFunCFLVs.