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A new class of codes, called product generator codes, which are similar to Elias's iterated codes, are investigated. An important subclass of these codes is the generalized Reed-Muller codes. If the original codes that are iterated to produce a product code are and -step orthogonalizable, then the product code is ( )-step orthogonalizable. Further if a th-order product generator code is produced from these original -step orthogonal izable codes, the new code is at most step orthogonalizable.