Error-locating codes--A new concept in error control

Abstract

A new coding technique is proposed lying midway between error-detection and error-correction coding. The block of received digits is regarded as subdivided into mutually exclusive sub-blocks. Errors occurring within particular sub-blocks are detected at the receiver and, in addition, the receiver is able to determine, by using the code redundancy, which particular sub-blocks contain errors. Such {\em error-locating} codes permit the location of digit errors to within a sub-block of the received message block without, in general, permitting the precise determination of erroneous digit positions. Two families of such codes are described, both of them limited to locating a single erroneous sub-block. One family permits the detection of up to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t-1</tex> errors in any one sub-block of length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</tex> . The second family locates up to two errors per sub-block, but is generally more efficient in its use of redundancy than the first family. Upper and lower bounds are given for the number of check digits required with any error-locating code. Codes meeting the lower bound exactly are termed {\em optimum} error-locating codes. All the codes of the second family, as well as some isolated examples of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t-1</tex> error-locating codes are optimum in this sense. The amount of redundancy required for such codes does not appear excessive and error location may provide an attractive alternative to conventional error detection in decision feedback communications.