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This paper presents a systematic procedure for finding optimum error-correcting group codes for the binary symmetric channel with m check digits and a minimum distance not less than do, where m and do are given integers. Some new schemes for reducing the computing time are used. The search procedure is readily programmable for computer execution and several programs were carried out on an IBM 7044. The newly found seven optimum triple-error-correcting group codes and six optimum double-error-correcting group codes including five quasi-perfect double-error-correcting codes are tabulated. Also, a list of optimum shortened cyclic codes found by a similar procedure is presented. The efficiency of the search procedure is demonstrated by the fact that the program yielded the new codes in a fairly short time.