New optimized Lcd codes and quantum codes using constacyclic codes over a non-local collection of rings Ak

In this article, we find several novel and efficient quantum error-correcting codes ( Q ecc ) by studying the structure of constacyclic ( C cc ), cyclic ( C c ), and negacyclic codes ( N C c ) over the ring A k = Z p r 1 , r 2 , ⋯ , r k / ⟨ ( r b ( m b + 1 ) - r b ) , r l r b = r b r l = 0 , b ≠ l ⟩ , where p = q m for m, m b ∈ N , m b | - 1 + q ∀ b , l ∈ 1 to k , q ≥ 3 is a prime, Z p is a finite field. We define distance-preserving gray map δ k . Moreover, we determine the quantum singleton defect ( Q SD) of Q ecc , which indicates their overall quality. We compare our codes with existing codes in recent publications. The rings discussed by Kong et al. (EPJ Quantum Technol 10:1–16, 2023), Suprijanto et al. (Quantum codes constructed from cyclic codes over the ring F q + vF q + v 2 F q + v 3 F q + v 4 F q , pp 1–14, 2021. arXiv: 2112.13488v2 [cs.IT] ), and Dinh et al. (IEEE Access 8:194082–194091, 2020) are specific cases of our work. Furthermore, we construct several novel and optimum linear complementary dual (Lcd) codes over A k .