On the existence of group codes for the Gaussian channel with odd

The question of the existence of nonplanar group codes for the Gaussian channel has been settled except in the case of odd and odd and composite. For this unsettled case, it is shown that the existence of a nonplanar group code is implied by the existence of a group satisfying i) is even, ii) has a faithful complex irreducible representation of the first kind, and iii) restricted to the two-Sylow subgroup of contains the identity representation. A partial converse of this existence result is also given. Finally, it is shown that for each odd not of the form , there exists a nonplanar group code with odd and composite.