Self-dual codes and quadratic residue codes over the ring ${\mathbb{Z}}_9+u{\mathbb{Z}}_9$

In this paper, we introduce a new definitions of the Gray weight and the Gray map for linear codes over ${\mathbb{Z}}_9+u{\mathbb{Z}}_9$ with $u^2=u$. Some results on self-dual codes over this ring are investigated. Further, the structural properties of quadratic residue codes are also considered. Two self-dual codes with parameters $[22,11,5]$ and $[24,12,9]$ over ${\mathbb{Z}}_9$ are obtained.