A generalization of Ding’s construction is proposed that employs as a defining set the collection of the sth powers ( $s\ge 2$ ) of all nonzero elements in $GF\left(p^m\right)$ , where $p\ge 2$ is prime. Some of the resulting codes are optimal or near-optimal and include projective codes over $GF\left(4\right)$ that give rise to optimal or near optimal quantum codes. In addition, the codes yield interesting combinatorial structures, such as strongly regular graphs and block designs.