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It is an interesting open question whether a self-dual quaternary code exists. It was shown by Conway and Pless that the only primes which can be orders of permutations in the group of are 11, 7, and 3. In this correspondence we eliminate 11 and 7 not only as permutations but also as orders of monomials in the group of . This is done by reducing the problems to the consideration of several codes and finding low weight vectors in these codes.