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Abstract: Additive codes over have been of great interest due to their application to quantum error correction. As another application, we introduce a new class of formally self-dual additive codes over , which is a natural analogue of the binary formally self-dual codes and is missing in the study of additive codes over . In fact, considered formally self-dual linear codes over of even lengths, and suggested classifying formally self-dual linear codes over of odd lengths in order to study lattices from these codes. These motivate our study on formally self-dual additive codes over . In this paper, we define extremal and near-extremal formally self-dual additive codes over , classify all extremal codes, and construct many near-extremal codes. We discuss a general method (called the weak balance principle) for constructing such codes. We conclude with some open problems. [Copyright &y& Elsevier]