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We present new classes of binary codes that are constructed on the basis of concatenated codes and product codes. We discuss the random-error-correction capabilities of these codes. Some examples of the codes for the correction of random errors are given which have at least as many codewords as the best codes previously known (to the authors) with the same minimum distance and same number of check symbols. The burst-error-correction capabilities of the codes are also discussed. Several examples of the codes for the correction of both random errors and burst errors are given. A decoding algorithm for the codes is also described.