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 ( R k ) 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 A Σ = { { 0 } , 2 S ⁎ , Σ , S ⁎ \ Σ } induces a translation scheme on ( S , + ) . We generalize these results by characterizing all supergroups Σ of T such that A Σ induces a symmetric translation scheme. In turn, we get new two-intersection sets in projective Hjelmslev geometries and two new series T q , k , s and U q , k , s of R-linear codes. The series T q , k , s generalizes the Teichmüller codes (special case s = 0 ). The codes U q , k , s are homogeneous two-weight codes. Application of the dualization construction to T q , k , s yields another series T q , k , s ⁎ . The Gray images of the codes T q , k , s and T q , k , s ⁎ have a higher minimum distance than all known F q -linear codes of the same length and size.