Encoding of quantum stabilizer codes over qudits with d = pk

Stabilizer codes over qudits have been widely studied but their encoding procedures have not been investigated in detail by many. The encoding procedure proposed previously by Grassi et al. generates the stabilizer codes over qudits for prime d, while it may only generate a subspace of the codespace over qudits where d is a power of a prime. In this paper, we propose an encoding procedure to generate the stabilizer codes over qudits for d being a power of a prime. The procedure involves reducing the problem of encoding over qudits with d = p(k) to a problem similar to encoding over qudits with d = p.