A family of single-error-correcting codes is described for the protection of binary data words of fixed lengthk, each of which has the same numberwof1's. The code family is shown to be valid for all integerskandw(where0 < w < k). Some other related codes, based upon conventional Hamming codes, Latin squares, and block designs are also developed, and some of these are more efficient for certain values ofwandk.

The run measure of a switching function has arisen in several contexts as an indication of the complexity, or cost, of a realization of the function. The run measure of a function can be defined in terms of its conventional truth-table representation. The output column of the truth table is an ordered sequence of zeros and ones that are disposed in runs; i.e., groups of like digits, of various lengths. The run measure of the function is simply the number of runs in this output sequence. It is often convenient to consider two functions to be equivalent if one can be obtained from the other by some permutation or complementation of the input variables. In this context, the cost of a function can be taken as the minimum value of the run measure over the equivalence class that contains the function. This paper derives a firm upper bound on the run measure for arbitrary n-variable switching functions when arbitrary permutations and complementations of the input variables are permitted. It is also shown that this bound is attained only in the case of the parity functions and that, hence in this sense, all other functions are less complex.

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 tot-1errors in any one sub-block of lengtht. 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 oft-1error-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.

A detailed study has been made of a certain class of systematic binary error-correcting codes that will correct the error bursts typical of some digital channels. These codes--generalizations of codes discovered by Abramson and Melas--are cyclic codes designed to correct any single burst of errors pern-digit word provided that the width of the burst (regarded cyclically) does not exceed a certain number of digits,b. Moreover, these codes are optimum in the sense that they employ the minimum number of redundant digits theoretically possible for a cyclic code with given values ofnandb. A cyclic code is completely characterized by its generator polynomialg(x), hence, the properties of the code can be determined by analysis of the correspondingg(x). Necessary and sufficient conditions ong(x)have been formulated for the corresponding cyclic code to be an optimum burst-bcorrecting code. These conditions have been formulated into a series of tests that can be carried out (in principle) on anyg(x). All optimum burst-bcyclic codes withn < 2^{12}andb < 6have been found in this way and their generators are tabulated in the paper. In all, 98 codes are listed (not counting reciprocals) forb = 3andb = 4; it was shown that no optimum codes exist forb = 5within the limits stated. Practical codes forb \geq 6will probably be nonoptimum codes because of the extreme word lengths required for optimum ones.

Binary group codes described by Abramson permit the correction of all single errors and all double errors in adjacent digits, with the use of significantly fewer check digits than codes capable of correcting all double-bit errors. This note considers the generalization of Abramson's codes to thep-nary case, where a symbol alphabet consisting of the digits0, 1, \cdots , p - 1is used for transmission,pbeing a prime number. Examples of suchp-nary codes are given, as well as necessary conditions for their existence. These codes bear the same relation to thep-nary Golay codes as Abramson's codes do to the familiar Hamming codes. Some as yet unanswered questions are raised, and suggestions for further possible generalizations are given.