Derivation of Quantum Codes from Specific Rings

Abstract

We study the construction of quantum error-correcting codes (QECCs) using the algebraic properties of cyclic codes defined over the product ring of the finite field of order p and a specific local chain ring. Our methodology relies on the analysis of Euclidean sums. This approach is used to construct generator polynomials and to calculate the dimension of these codes. A key part of our approach is the use of an isometric Gray mapping. Our technique preserves distances while adapting these classical structures to quantum systems. As a result, it generates new families of good quantum codes.