Error-Correcting Code
An error-correcting code is an algorithm for expressing a sequence of numbers such that any errors which are introduced canbe detected and corrected (within certain limitations) based on the remaining numbers. The study of error-correctingcodes and the associated mathematics is known as Coding Theory.
Error detection is much simpler than errorcorrection, 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 useconvolution codes. Error-correcting codes are also used in CD players, high speed modems, and cellular phones. Modemsuse error detection when they compute Checksums, which are sums of the digits in a given transmissionmodulo 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 ofDigits 
. Take 
and double. Now add the number ofDigits in Odd positions which are 
to this number. Now add 
. The checknumber is then the number required to bring the last Digit to 0. This scheme detects all single Digit errorsand 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.
- Mice Problem
- Mid-Arc Points
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- Milin Conjecture
- Mill
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