Codes from translation schemes on Galois rings of characteristic 4

Titelangaben

Kiermaier, Michael:
Codes from translation schemes on Galois rings of characteristic 4.
In: Electronic Notes in Discrete Mathematics. Bd. 40 (2013) . - S. 175-180.
ISSN 1571-0653
DOI: https://doi.org/10.1016/j.endm.2013.05.032

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Abstract

In [4], 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 [8, Fact 5.2]. The resulting Teichmüller Codes have a high minimum distance.

The key step in the proof of the two-weight property in [4] is to show that for a certain supergroup In [4], 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 [8, Fact5.2]. The resulting Teichmüller Codes have a high minimum distance. of the Teichmüller units T in a Galois ring S of characteristic 4, the partition A_Σ = {{0},2S^*, 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.