A linear coding scheme for the correction of all possible single errors during arithmetic operations on binary number representations is discussed. The coded form of a numberkis the binary representation of the numberkt, wheretis a positive integer. The code corrects all errors of the form\pm2^i, whereiis less thann, the number of digits in the code words. The problem of determining the largest numberkwhich may be encoded for a particular value oftis discussed. It is also shown that as the number of arithmetic operations to be performed increases, the use of this coding scheme becomes more significant in improving the reliability of the computational unit. An asymmetric process (in which only 1-errors or only O-errors are to be corrected) is also investigated and compared with the symmetric process.

In a highly-asymmetric binary channel it may be necessary to correct only those errors which result from incorrect transmission of one of the two code elements. Minimum weight-distance relationships and rules for generating single-error correcting codes in such situations are given. More code characters are generally obtained for a given character length than are obtained with codes designed for single-error correction in symmetric channels. Examples are given, including one which specifies the code which results in the highest average probability of correct transmission of equiprobable messages through a highly-asymmetric channel.

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 correctsk-tuple 1-errors, detects(k+a)-tuple 1-errors, correctsj-tuple 0-errors, and detects(j+b)-tuple O-errors(k,a,j, andb, are non-negative integers withk > janda \geq b). These requirements are weaker than those for a symmetricalk-tuple error correcting,(k+a)-tuple error detecting, code and hence may be used to generally obtain more code characters for a given character length than are obtainable in thek, (k+a)-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 fork-tuple 1-error correction which also yield more code characters than symmetricalk-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(k+l)^{-1}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.