Quantum Cyclic Codes Over ℤ m $$ {\mathbb{Z}}_m $$

Abstract

Quantum codes over finite rings have the advantage of being able to adapt to quantum physical systems with arbitrary order. Furthermore, operations are much easier to execute in finite rings than they are in fields. This paper discusses quantum cyclic codes over the modulo m residue class ring $$ {\mathbb{Z}}_m $$ . A connection is established between the stabilizer codes over $$ {\mathbb{Z}}_m $$ and the additive codes over an extension ring of $$ {\mathbb{Z}}_m $$ that generalizes the well-known relationship between the stabilizer codes over $$ {\mathbb{F}}_q $$ and the additive codes over $$ {\mathbb{F}}_{q^2} $$ . We prove that if the irreducible polynomial is selected according to a simple criterion, the additive codes which are self-orthogonal with respect to the conjugate inner product correspond to the stabilizer codes. The structure of cyclic stabilizer codes is developed, and some simple conditions for finding them are presented. We also define the quantum Bose-Chaudhuri-Hocquenghem (BCH) and quantum Reed-Solomon (RS) codes over $$ {\mathbb{Z}}_m $$ . Finally, new quantum cyclic codes over $$ {\mathbb{Z}}_m $$ are given.