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A relevant problem pertaining to the theory of runs is considered. The solution is given, and in the sequel, a useful identity (Lemma 2) is derived. It is shown how these results apply to decoding of systematic cyclic codes. This leads to a simply implemented error-correcting and detecting decoder. The decoder functions by searching for an error-free string of consecutive digits. The efficiency of such a decoder is described. The quantitative values are given in Table I. The decoding efficiency is higher when errors occur in bursts, instead of being independently distributed. The use of feedback offers an attractive utilization of the intrinsic error-detecting capability.