An error-correcting code is an algorithm for expressing a sequence of numbers such that any errors which are introduced can be detected and corrected (within certain limitations) based on the remaining numbers. The study of error-correcting codes and the associated mathematics is known as Coding Theory.

Error detection is much simpler than error correction, and one or more ``check'' digits are commonly embedded in credit card numbers in order to detect mistakes. Early space probes like Mariner used a type of error-correcting code called a block code, and more recent space probes use convolution codes. Error-correcting codes are also used in CD players, high speed modems, and cellular phones. Modems use error detection when they compute Checksums, which are sums of the digits in a given transmission modulo some number. The ISBN used to identify books also incorporates a check Digit.

A powerful check for 13 Digit numbers consists of the following. Write the number as a string of Digits . Take and double. Now add the number of Digits in Odd positions which are to this number. Now add . The check number is then the number required to bring the last Digit to 0. This scheme detects all single Digit errors and all Transpositions of adjacent Digits except 0 and 9.

**References**

Conway, J. H. and Sloane, N. J. A. ``Error-Correcting Codes.''
§3.2 in *Sphere Packings, Lattices, and Groups, 2nd ed.*
New York: Springer-Verlag, pp. 75-88, 1993.

Gallian, J. ``How Computers Can Read and Correct ID Numbers.'' *Math Horizons*, pp. 14-15, Winter 1993.

Guy, R. K. *Unsolved Problems in Number Theory, 2nd ed.* New York: Springer-Verlag,
pp. 119-121, 1994.

MacWilliams, F. J. and Sloane, N. J. A. *The Theory of Error-Correcting Codes.* Amsterdam, Netherlands: North-Holland, 1977.

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1999-05-25