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The final character of a ten-digit International Standard Book Number is a check digit computed so that multiplying each digit by its position in the number (counting from the right) and taking the sum of these products modulo 11 is 0. The digit the farthest to the right (which is multiplied by 1) is the check digit, chosen to make the sum correct.
The check digit is computed as follows: Drop the check digit from the number (if it's already present). This leaves the payload. Start with the payload digits. Moving from right to left, double every second digit, starting from the last digit. If doubling a digit results in a value > 9, subtract 9 from it (or sum its digits).
The validity of a digit sequence containing a check digit is defined over a quasigroup. A quasigroup table ready for use can be taken from Damm's dissertation (pages 98, 106, 111). [3] It is useful if each main diagonal entry is 0, [1] because it simplifies the check digit calculation.
Verhoeff had the goal of finding a decimal code—one where the check digit is a single decimal digit—which detected all single-digit errors and all transpositions of adjacent digits. At the time, supposed proofs of the nonexistence [6] of these codes made base-11 codes popular, for example in the ISBN check digit.
The last digit is the Luhn check digit. IINs and PANs have a certain level of internal structure and share a common numbering scheme set by ISO/IEC 7812. The parts of the number are as follows: a six or eight-digit Issuer Identification Number (IIN), [a] the first digit of which is the major industry identifier (MII)
A digit (in a given position in the number) that is lower than its corresponding threshold value means that it is the most-significant digit, hence in the string this is the end of the number, and the next symbol (if present) is the least-significant digit of the next number.
The result is appended to the message as an extra word. In simpler terms, for n =1 this means adding a bit to the end of the data bits to guarantee that there is an even number of '1's. To check the integrity of a message, the receiver computes the bitwise exclusive or of all its words, including the checksum; if the result is not a word ...
Code 11 is a barcode symbology developed by Intermec in 1977, and it is used primarily in telecommunications. The symbol can encode any length string consisting of the digits 0–9 and the dash character (-). A twelfth code represents the start/stop character, commonly printed as "*". One or two modulo-11 check digit(s) can be included.