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Damm algorithm
In error detection, the Damm algorithm is a check digit algorithm that detects all single-digit errors and all adjacent transposition errors. It was presented by H. Michael Damm in 2004, as a part of his PhD dissertation entitled Totally Antisymmetric Quasigroups.
The Damm algorithm is similar to the Verhoeff algorithm. It too will detect all occurrences of the two most frequently appearing types of transcription errors, namely altering a single digit or transposing two adjacent digits (including the transposition of the trailing check digit and the preceding digit). The Damm algorithm has the benefit that it does not have the dedicatedly constructed permutations and its position-specific powers of the Verhoeff scheme. A table of inverses can also be dispensed with when all main diagonal entries of the operation table are zero.
The Damm algorithm generates only 10 possible values, avoiding the need for a non-digit character (such as the X in the 10-digit ISBN check digit scheme).
Prepending leading zeros does not affect the check digit (a weakness for variable-length codes).
There are totally anti-symmetric quasigroups that detect all phonetic errors associated with the English language (13 ↔ 30, 14 ↔ 40, ..., 19 ↔ 90). The table used in the illustrating example is based on an instance of such kind.
For all checksum algorithms, including the Damm algorithm, prepending leading zeroes does not affect the check digit, so 1, 01, 001, etc. produce the same check digit. Consequently variable-length codes should not be verified together.
Its essential part is a quasigroup of order 10 (i.e. having a 10 × 10 Latin square as the body of its operation table) with the special feature of being weakly totally anti-symmetric. Damm revealed several methods to create totally anti-symmetric quasigroups of order 10 and gave some examples in his doctoral dissertation. With this, Damm also disproved an old conjecture that totally anti-symmetric quasigroups of order 10 do not exist.
A quasigroup (Q, ∗) is called totally anti-symmetric if for all c, x, y ∈ Q, the following implications hold:
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Damm algorithm
In error detection, the Damm algorithm is a check digit algorithm that detects all single-digit errors and all adjacent transposition errors. It was presented by H. Michael Damm in 2004, as a part of his PhD dissertation entitled Totally Antisymmetric Quasigroups.
The Damm algorithm is similar to the Verhoeff algorithm. It too will detect all occurrences of the two most frequently appearing types of transcription errors, namely altering a single digit or transposing two adjacent digits (including the transposition of the trailing check digit and the preceding digit). The Damm algorithm has the benefit that it does not have the dedicatedly constructed permutations and its position-specific powers of the Verhoeff scheme. A table of inverses can also be dispensed with when all main diagonal entries of the operation table are zero.
The Damm algorithm generates only 10 possible values, avoiding the need for a non-digit character (such as the X in the 10-digit ISBN check digit scheme).
Prepending leading zeros does not affect the check digit (a weakness for variable-length codes).
There are totally anti-symmetric quasigroups that detect all phonetic errors associated with the English language (13 ↔ 30, 14 ↔ 40, ..., 19 ↔ 90). The table used in the illustrating example is based on an instance of such kind.
For all checksum algorithms, including the Damm algorithm, prepending leading zeroes does not affect the check digit, so 1, 01, 001, etc. produce the same check digit. Consequently variable-length codes should not be verified together.
Its essential part is a quasigroup of order 10 (i.e. having a 10 × 10 Latin square as the body of its operation table) with the special feature of being weakly totally anti-symmetric. Damm revealed several methods to create totally anti-symmetric quasigroups of order 10 and gave some examples in his doctoral dissertation. With this, Damm also disproved an old conjecture that totally anti-symmetric quasigroups of order 10 do not exist.
A quasigroup (Q, ∗) is called totally anti-symmetric if for all c, x, y ∈ Q, the following implications hold: