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The
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Verhoeff's Dihedral Group D5 Check
The mathematical description of this technique may appear complex but in practice it can be reduced to a pair of two-dimensional arrays, a single dimensional inverse array and a simple computational procedure. These three arrays are shown in the following tables.
- The first array contains the result of "Dihedral D5" multiplication;
- The second array consists of 8 rows of which two are defined while the rest are derived by applying the following formula: F(i, j) = F(i - 1, F(1, j)) ;
- The third array consists of a single row containing the inverse of the Dihedral D5 array it identifies the location of all the zero values in the first array.
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Results of Dihedral D5 multiplication |
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
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0 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
1 | 1 | 2 | 3 | 4 | 0 | 6 | 7 | 8 | 9 | 5 |
2 | 2 | 3 | 4 | 0 | 1 | 7 | 8 | 9 | 5 | 6 |
3 | 3 | 4 | 0 | 1 | 2 | 8 | 9 | 5 | 6 | 7 |
4 | 4 | 0 | 1 | 2 | 3 | 9 | 5 | 6 | 7 | 8 |
5 | 5 | 9 | 8 | 7 | 6 | 0 | 4 | 3 | 2 | 1 |
6 | 6 | 5 | 9 | 8 | 7 | 1 | 0 | 4 | 3 | 2 |
7 | 7 | 6 | 5 | 9 | 8 | 2 | 1 | 0 | 4 | 3 |
8 | 8 | 7 | 6 | 5 | 9 | 3 | 2 | 1 | 0 | 4 |
9 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
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The full array for Function F |
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
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0 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
1 | 1 | 5 | 7 | 6 | 2 | 8 | 3 | 0 | 9 | 4 |
2 | 5 | 8 | 0 | 3 | 7 | 9 | 6 | 1 | 4 | 2 |
3 | 8 | 9 | 1 | 6 | 0 | 4 | 3 | 5 | 2 | 7 |
4 | 9 | 4 | 5 | 3 | 1 | 2 | 6 | 8 | 7 | 0 |
5 | 4 | 2 | 8 | 6 | 5 | 7 | 3 | 9 | 0 | 1 |
6 | 2 | 7 | 9 | 3 | 8 | 0 | 6 | 4 | 1 | 5 |
7 | 7 | 0 | 4 | 6 | 9 | 1 | 3 | 2 | 5 | 8 |
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The Inverse D5 array |
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
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0 | 4 | 3 | 2 | 1 | 5 | 6 | 7 | 8 | 9 |
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The identifier is checked by starting at the rightmost digit of the identifier (the check-digit itself) and proceeding to the left processing each digit as follows:
- Check = ArrayDihedralD5 ( Check, ArrayFunctionF(( Position Modulus 8), Digit ))
Check = the running value of the check-sum (starts at zero and modified by each step).
Position = the position of the digit (counted from the right starting at zero).
Digit = the value of the digit.
The final value of Check should be zero. Otherwise the check has failed.
When calculating the
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Position is the position that the digit will have when the
has been appended.Gloss PreSpace false t check-digit The final value of Check is applied to the Inverse D5 array to find the correct
.Gloss PreSpace false t check-digit
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Sample Java Script for computing Verhoeff's Dihedral Check
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A live version of an HTML form and JavaScript is available in section 6.4.1 SNOMED CT Identifier Check. |
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<style> p.p1 {margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica} span.s1 {color: #021da7} span.s2 {color: #f9975e} span.s3 {color: #ff9450} span.s4 {color: #ab4500} span.s5 {color: #a7a400} table {border-width: 6px; border-color: #0080ff; border-collapse: collapse; border-style: ridge;} td {border-width: 3px; border-color: #0080ff; border-collapse: collapse; padding: 6px; border-style: ridge;} </style> <form action="" name="form"> <table width="441"> <tr> <td width="212" height="25"> Partial Identifier <br/>(without check-digit) </td> <td width="115" height="25"> <input name="num" size="18"/> </td> <td width="92" height="25"> <input onclick="VerhoeffCompute()" type="button" value="Compute"/> </td> </tr> <tr> <td width="212" height="35"> SNOMED CT Identifier </td> <td width="115" height="35"> <input name="numcd" size="18"/> </td> <td width="92" height="35"> <input onclick="VerhoeffCheck()" type="button" value="Check"/> </td> </tr> <tr> <td width="212" height="23"> Result of check </td> <td width="115" height="23" colspan="2" id="out"> </td> </tr> <tr> <td width="212" height="23"> Component type </td> <td width="115" height="23" colspan="2" id="component"> </td> </tr> <tr> <td width="212" height="23"> Namespace </td> <td width="115" height="23" colspan="2" id="extnamespace"> </td> </tr> </table> <p style="margin-left: 0; margin-right: 0"> This Verhoeff checking part of this code was based on a webpage at: </p> <ul> <li> <a href="http://www.augustana.ab.ca/~mohrj/algorithms/checkdigit.html"> http://www.augustana.ab.ca/~mohrj/algorithms/checkdigit.html </a> </li> </ul> </form> |
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var FnF = new Array(); FnF[0] = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]; FnF[1] = [1, 5, 7, 6, 2, 8, 3, 0, 9, 4]; for ( var i = 2; i < 8; i++ ) { FnF[i] = [,,,,,,,,,]; for ( var j = 0; j < 10; j++ ) FnF[i][j] = FnF[i - 1][FnF[1][j]]; } var Dihedral = new Array( [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 0, 6, 7, 8, 9, 5], [2, 3, 4, 0, 1, 7, 8, 9, 5, 6], [3, 4, 0, 1, 2, 8, 9, 5, 6, 7], [4, 0, 1, 2, 3, 9, 5, 6, 7, 8], [5, 9, 8, 7, 6, 0, 4, 3, 2, 1], [6, 5, 9, 8, 7, 1, 0, 4, 3, 2], [7, 6, 5, 9, 8, 2, 1, 0, 4, 3], [8, 7, 6, 5, 9, 3, 2, 1, 0, 4], [9, 8, 7, 6, 5, 4, 3, 2, 1, 0] ); var InverseD5 = new Array(0, 4, 3, 2, 1, 5, 6, 7, 8, 9 ); function VerhoeffCheck() { var check = 0; var IdValue = document.form.numcd.value; document.getElementById("out").innerText = ""; document.getElementById("out").setAttribute("style","color:red;"); document.getElementById("component").innerText ="Invalid partition"; document.getElementById("component").setAttribute("style","color:green;"); document.getElementById("extnamespace").innerText ="No namespace"; document.getElementById("extnamespace").setAttribute("style","color:red;"); for ( var i=IdValue.length-1; i >=0; i-- ) check = Dihedral[check][FnF[(IdValue.length-i-1) % 8][IdValue.charAt(i)]]; if ( check != 0 ) { document.getElementById("out").innerText = "Check-digit ERROR"; } else if (IdValue.length < 6) {document.getElementById("out").innerText = "SCTID too short";} else if (IdValue.length > 18) {document.getElementById("out").innerText = "SCTID too long";} else {document.getElementById("out").innerText = "Check-digit OK"; document.getElementById("out").setAttribute("style","color:green;"); switch (IdValue.substr(IdValue.length-3,2)) { case "00": document.getElementById("component").innerText ="Concept"; document.getElementById("extnamespace").innerText ="International"; break; case "01": document.getElementById("component").innerText ="Description"; document.getElementById("extnamespace").innerText ="International"; break; case "02": document.getElementById("component").innerText ="Relationship"; document.getElementById("extnamespace").innerText ="International"; break; case "03": document.getElementById("component").innerText ="Subset (RF1)"; document.getElementById("extnamespace").innerText ="International"; break; case "04": document.getElementById("component").innerText ="Cross Map Set (RF1)"; document.getElementById("extnamespace").innerText ="International"; break; case "05": document.getElementById("component").innerText ="Cross Map Target (RF1)"; document.getElementById("extnamespace").innerText ="International"; break; case "10": document.getElementById("component").innerText ="Concept"; document.getElementById("extnamespace").innerText =IdValue.substr(IdValue.length-10,7); break; case "11": document.getElementById("component").innerText ="Description"; document.getElementById("extnamespace").innerText =IdValue.substr(IdValue.length-10,7); break; case "12": document.getElementById("component").innerText ="Relationship"; document.getElementById("extnamespace").innerText =IdValue.substr(IdValue.length-10,7); break; case "13": document.getElementById("component").innerText ="Subset (RF1)"; document.getElementById("extnamespace").innerText =IdValue.substr(IdValue.length-10,7); break; case "14": document.getElementById("component").innerText ="Cross Map Set (RF1)"; document.getElementById("extnamespace").innerText =IdValue.substr(IdValue.length-10,7); break; case "15": document.getElementById("component").innerText ="Cross Map Target (RF1)"; document.getElementById("extnamespace").innerText =IdValue.substr(IdValue.length-10,7); break; default: document.getElementById("component").setAttribute("style","color:red;"); } if (document.getElementById("extnamespace").innerText=='International') {document.getElementById("extnamespace").setAttribute("style","color:green;");} else if (IdValue.length>10) {document.getElementById("extnamespace").setAttribute("style","color:green;");} else {document.getElementById("extnamespace").innerText="Invalid Namespace"; } } } function VerhoeffCompute( ) { var IdValue = document.form.num.value; var check = 0; document.form.numcd.value= ""; for ( var i = IdValue.length-1; i >=0; i-- ) check = Dihedral[check][FnF[(IdValue.length-i) % 8][IdValue.charAt(i)]]; document.form.numcd.value = document.form.num.value + InverseD5[check]; VerhoeffCheck(); document.getElementById("out").innerText = "Computed check-digit"; } |
Sample Visual Basic for computing Verhoeff's Dihedral Check
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Option Explicit Private Dihedral(9) As Variant Private FnF(7) As Variant Private InverseD5 As Variant Public Function VerhoeffCheck(ByVal IdValue As String) As Boolean 'Check the supplied value and return true or false Dim tCheck As Integer, i As Integer VerhoeffArrayInit For i = Len(IdValue) To 1 Step -1 tCheck = Dihedral(tCheck)(FnF((Len(IdValue) - i) Mod 8)(Val(Mid(IdValue, i, 1)))) Next VerhoeffCheck = tCheck = 0 End Function Public Function VerhoeffCompute(ByVal IdValue As String) As String 'Compute the check digit and return the identifier complete with check-digit Dim tCheck As Integer, i As Integer VerhoeffArrayInit For i = Len(IdValue) To 1 Step -1 tCheck = Dihedral(tCheck)(FnF((Len(IdValue) - i + 1) Mod 8)(Val(Mid(IdValue, i, 1)))) Next VerhoeffCompute = IdValue & InverseD5(tCheck) End Function Private Sub VerhoeffArrayInit() 'Create the arrays required Dim i As Integer, j As Integer 'if already created exit here If VarType(InverseD5) >= vbArray Then Exit Sub 'create the DihedralD5 array Dihedral(0) = Array(0, 1, 2, 3, 4, 5, 6, 7, 8, 9) Dihedral(1) = Array(1, 2, 3, 4, 0, 6, 7, 8, 9, 5) Dihedral(2) = Array(2, 3, 4, 0, 1, 7, 8, 9, 5, 6) Dihedral(3) = Array(3, 4, 0, 1, 2, 8, 9, 5, 6, 7) Dihedral(4) = Array(4, 0, 1, 2, 3, 9, 5, 6, 7, 8) Dihedral(5) = Array(5, 9, 8, 7, 6, 0, 4, 3, 2, 1) Dihedral(6) = Array(6, 5, 9, 8, 7, 1, 0, 4, 3, 2) Dihedral(7) = Array(7, 6, 5, 9, 8, 2, 1, 0, 4, 3) Dihedral(8) = Array(8, 7, 6, 5, 9, 3, 2, 1, 0, 4) Dihedral(9) = Array(9, 8, 7, 6, 5, 4, 3, 2, 1, 0) 'create the FunctionF array FnF(0) = Array(0, 1, 2, 3, 4, 5, 6, 7, 8, 9) FnF(1) = Array(1, 5, 7, 6, 2, 8, 3, 0, 9, 4) 'compute the rest of the FunctionF array For i = 2 To 7 FnF (i) = Array(0, 0, 0, 0, 0, 0, 0, 0, 0, 0) For j = 0 To 9 FnF (i)(j) = FnF(i - 1)(FnF(1)(j)) Next Next 'Create the InverseD5 array InverseD5 = Array("0", "4", "3", "2", "1", "5", "6", "7", "8", "9") End Sub |
Reasons for using a check-digit
Although a user should rarely type the
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- single errors: a becomes b (60% to 95% of all errors).
- omitting or adding a digit (10% to 20%).
- adjacent transpositions: ab becomes ba (10% to 20%).
- twin errors: aa becomes bb (0.5% to 1.5%).
- jump transpositions: acb becomes bca (0.5% to 1.5%).
- jump twin errors: aca becomes bcb (below 1%).
- phonetic errors: a0 becomes 1a -similar pronunciation e.g. thirty or thirteen (0.5% to 1.5%).
In the explanations above, a is not equal to b, but c can be any decimal digit.
A brief comparison of check-digit effectiveness
The IBM Check
The check-sums used for credit cards (the IBM check) picks up the most common errors but miss some adjacent transpositions and many jump transpositions. Assuming the pattern of errors described above, on average it will miss between 4% and 5% of expected errors.
The ISBN Check (Modulus 11)
The ISBN modulus 11 (used for
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Partitions could be a useful way of distinguishing developmental and released components and revising the partition and recalculating the
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Verhoeff's Check
Verhoeff's check catches all single errors, all adjacent transpositions, over 95% of twin errors, over 94% of jump transpositions and jump twin errors, and most phonetic errors. Therefore, like modulus 11, the Verhoeff check reduces the undetected error rate to 2% or 3%. Unlike modulus 11, it does this using a single decimal
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The majority of the undetected errors with both modulus 11 and Verhoeff result from additions or omissions of digits. Any
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