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A class of product codes for which an augmented syndrome can be defined is studied. By using this augmented syndrome, we are able to add new codewords to a product code while keeping the minimum distance fixed. In many cases the resulting code is as good as any known code. Decoders that may be simpler to implement than existing decoders are described for these augmented codes. It is also seen that in certain cases, by choosing not to correct a small number of correctable errors, significant simplifications in the decoder can be achieved. Finally, this augmentation technique is used in an example to generate a code of length 32 and minimum distance 8 with more codewords than any code previously known.