The Gilbert-Varshamov Bound for Stabilizer Codes Over <inline-formula> <tex-math notation="LaTeX">$\mathbb{Z}_m$ </tex-math> </inline-formula>

Abstract

Quantum codes over finite rings have received a great deal of attention in recent years. Compared with quantum codes over finite fields, a notable advantage of quantum codes over finite rings is that they can adapt to quantum physical systems of arbitrary order. Moreover, operations are much easier to execute in finite rings than they are in fields. The modulo $m$ residue class ring $\mathbb {Z}_{m}$ is the most common finite ring. This paper investigates stabilizer codes over $\mathbb {Z}_{m}$ and presents the Gilbert–Varshamov (GV) bound. The GV bound shows that surprisingly good quantum codes exist over $\mathbb {Z}_{m}$ , which makes quantum coding feasible for arbitrary quantum physical system. We also provide an enhanced version of the GV bound for non-degenerate stabilizer codes over $\mathbb {Z}_{m}$ . The enhanced GV bound has an asymptotical form and ensures the existence of asymptotically good stabilizer codes over $\mathbb {Z}_{m}$ . Finally, these two bounds are well suited for computer searching.