Construction of MDS self-dual codes over Galois rings

The purpose of this paper is to construct nontrivial MDS self-dual codes over Galois rings. We consider a building-up construction of self-dual codes over Galois rings as a GF(q)-analogue of [20]. We give a necessary and sufficient condition on which the building-up construction holds. We construct MDS self-dual codes of lengths up to 8 over GR(3, 2), GR(3, 2) and GR(3, 2), and near-MDS self-dual codes of length 10 over these rings. In a similar manner, over GR(5, 2), GR(5, 2) and GR(7, 2), we construct MDS self-dual codes of lengths up to 10 and near-MDS self-dual codes of length 12. Furthermore, over GR(11, 2) we have MDS self-dual codes of lengths up to 12.