Construction of Quantum Codes from Constacyclic Codes Over the Class of Commutative Rings

Abstract

This paper primarily investigates the structural properties of constacyclic codes over the ring  , defined as  where i, j = 1, 2, 3, ..., k, and k, m are positive integers. Here,  denotes a finite field of order p^m with characteristic p, an odd prime. Furthermore, we determine the necessary and sufficient conditions for the duals of constacyclic codes to exist. These findings make it easier to create new Quantum Error-Correcting (QEC) codes throughout the ring  (i.e., when k = 1), as well as optimal linear codes that make use of the Gray images of constacyclic codes. Additionally, Table 4 presents several Linear Complementary Dual (LCD) codes obtained using the Gray map.