Codes from translation schemes on Galois rings of characteristic 4

In [Thomas Honold. Two-intersection sets in projective Hjelmslev spaces. In Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems, pages 1807–1813, 2010], it has been shown that the Teichmüller point set in the projective Hjelmslev geometry PHG(Rk) over a Galois ring R of characteristic 4 with k odd is a two-intersection set. From this result, the parameters of the generated codes can be derived, see [Michael Kiermaier and Johannes Zwanzger. New ring-linear codes from dualization in projective Hjelmslev geometries. To appear in Des. Codes Cryptogr. doi: 10.1007/s10623-012-9650-1, Fact 5.2]. The resulting Teichmüller Codes have a high minimum distance. The key step in the proof of the two-weight property in [Thomas Honold. Two-intersection sets in projective Hjelmslev spaces. In Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems, pages 1807–1813, 2010.] is to show that for a certain supergroup Σ of the Teichmüller units T in a Galois ring S of characteristic 4, the partition AΣ={{0},2S⁎,Σ,S⁎\Σ} induces a translation scheme on (S,+). We generalize these results by characterizing all supergroups Σ of T such that AΣ induces a symmetric translation scheme. In turn, we get new two-intersection sets in projective Hjelmslev geometries and two new series Tq,k,s and Uq,k,s of R-linear codes. The series Tq,k,s generalizes the Teichmüller codes (special case s=0). The codes Uq,k,s are homogeneous two-weight codes. Application of the dualization construction to Tq,k,s yields another series . The Gray images of the codes Tq,k,s and have a higher minimum distance than all known Fq-linear codes of the same length and size.