Skew codes over the split quaternions

Abstract

In this paper, the structures of linear codes over the split quaternions with coefficients from $Z_3, Hs, 3 = Z_3 +iZ_3 +jZ_3 + kZ_3$ are determined with $i^2 = −1, j^2 = k^2 = 1, ij = k = −ji, jk = −i = −kj, ki = j = −ik, ijk = 1$. It is shown that the split quaternions over $Z_3$ decompose into two parts from $Z_3+iZ_3$ with idempotent coefficients. The structures of the skew cyclic and skew constacyclic codes over $Hs,3$ are obtained.