We study the algebraic structures of constacyclic codes over a new finite non-chain ring R and their l-Galois dual codes based on a new Gray map ϕ and determine the dimensions of the l-Galois hull of constacyclic codes over R. Furthermore, we propose a quantum construction X of the Galois inner product to construct quantum codes by combining the Calderbank–Shor–Steane (CSS) construction, Hermitian construction and construction X. As an application, by calculating the dimensions of the l-Galois hull of constacyclic codes over R, we obtain some new quantum maximum distance separable (QMDS) codes and new quantum error-correcting codes (QECCs) that are better than the best known ones.
Zhang et al. (2026) studied this question.