ATRACTOR

Firstly, it follows from the permutation considered and the operation of the dihedral group \(D_{5}\), that this error-detecting code detects all singular errors. And adjacent transpositions? Let us analyse the case of the two rightmost digits \(x_{7}\) and \(x_{8}\) of the identification number. The following table shows the result of operating \(x_{7}\) (after the permutation) with \(x_{8}\), in the dihedral group \(D_{5}\). In brackets, you may see the result when the two digits are swapped.

		\[x_{8}\]
		0	1	2	3	4	5	6	7	8	9
\(x_{7}\)	0	0 (0)	1 (4)	2 (3)	3 (2)	4 (1)	5 (8)	6 (9)	7 (5)	8 (6)	9 (7)
	1	4 (1)	0 (0)	1 (4)	2 (3)	3 (2)	9 (7)	5 (8)	6 (9)	7 (5)	8 (6)
	2	3 (2)	4 (1)	0 (0)	1 (4)	2 (3)	8 (6)	9 (7)	5 (8)	6 (9)	7 (5)
	3	2 (3)	3 (2)	4 (1)	0 (0)	1 (4)	7 (5)	8 (6)	9 (7)	5 (8)	6 (9)
	4	1 (4)	2 (3)	3 (2)	4 (1)	0 (0)	6 (9)	7 (5)	8 (6)	9 (7)	5 (8)
	5	8 (5)	7 (9)	6 (8)	5 (7)	9 (6)	3 (3)	2 (4)	1 (0)	0 (1)	4 (2)
	6	9 (6)	8 (5)	7 (9)	6 (8)	5 (7)	4 (2)	3 (3)	2 (4)	1 (0)	0 (1)
	7	5 (7)	9 (6)	8 (5)	7 (9)	6 (8)	0 (1)	4 (2)	3 (3)	2 (4)	1 (0)
	8	6 (8)	5 (7)	9 (6)	8 (5)	7 (9)	1 (0)	0 (1)	4 (2)	3 (3)	2 (4)
	9	7 (9)	6 (8)	5 (7)	9 (6)	8 (5)	2 (4)	1 (0)	0 (1)	4 (2)	3 (3)

As you may see, the system detects all possible transpositions between \(x_{7}\) and \(x_{8}\). Likewise, the system will detect all the other transpositions between any pair of consecutive digits: check it here. You may check also, for instance, if the system detects all intercalated transpositions; You will see that it does not (just because of the given fixed permutation that defines the system; but it is possible to replace this permutation by another one that will make the system also 100% effective in detecting all intercalated transpositions).

In conclusion, this system is 100% effective in detecting the two most common errors: singular and adjacent transpositions.