Abstract
In an asymmetric binary channel, it may be sufficient to correct single O-errors and detect double 0-errors, for example, while correcting double 1-errors and detecting quadruple 1-errors. (A double 1-error is said to occur when two of the l's of an input code character are delivered as O's at the output of the channel.) Minimum distance requirements are given for pairs of code characters of a code which corrects <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> -tuple 1-errors, detects <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(k+a)</tex> -tuple 1-errors, corrects <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</tex> -tuple 0-errors, and detects <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(j+b)</tex> -tuple O-errors <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(k,a,j</tex> , and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</tex> , are non-negative integers with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k > j</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a \geq b</tex> ). These requirements are weaker than those for a symmetrical <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> -tuple error correcting, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(k+a)</tex> -tuple error detecting, code and hence may be used to generally obtain more code characters for a given character length than are obtainable in the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k, (k+a)</tex> -case. If the channel is highly asymmetric, it may be sufficient to detect and correct only one type of error. An earlier paper considered the case of single 1-error correction and showed that it was always possible to obtain more code characters than exist in known single error correcting codes of equivalent character length except in cases where the symmetric code is "close-packed." In this paper codes are developed for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> -tuple 1-error correction which also yield more code characters than symmetrical <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> -tuple error correcting codes of the same length. The correction scheme is generally symbol-correcting, but may require message-correction of binary sequences whose length is approximately <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(k+l)^{-1}</tex> that of the code characters. A double 1-error correcting code is discussed in some detail and examples of code generation and correction are included.
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