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The attractiveness of majority-logic decoding is its simple implementation. Several classes of majority-logic decodable block codes have been discovered for the past two decades. In this paper, a method of constructing a new class of majority-logic decodable block codes is presented. Each code in this class is formed by combining majority-logic decodable codes of shorter lengths. A procedure for orthogonalizing codes of this class is formulated. For each code, a lower bound on the number of correctable errors with majority-logic decoding is obtained. An upper bound on the number of orthogonalization steps for decoding each code is derived. Several majority-logic decodable codes that have more information digits than the Reed-Muller codes of the same length and the same minimum distance are found. Some results presented in this paper are extensions of the results of Lin and Weldon  and Gore  on the majority-logic decoding of direct product codes.