Hermitian hull of constacyclic codes over a class of non-chain rings and new quantum codes.

Let p be a prime number and q = p m for some positive integer m. In this paper, we find the possible Hermitian hull dimensions of λ -constacyclic codes over R e = F q 2 + u F q 2 + u 2 F q 2 + ⋯ + u e - 1 F q 2 , u e = 1 where F q 2 is the finite field of q 2 elements, e | (q + 1) and λ = η 1 α 1 + η 2 α 2 + ⋯ + η e α e for α l ∈ F q 2 ∗ of order r l such that r l ∣ q + 1 (for each 1 ≤ l ≤ e ). Further, we obtain some conditions for these codes to be Hermitian LCD. Also, under certain conditions, we establish a strong result that converts every constacyclic code to a Hermitian LCD code (Corollaries 2 and 3). We also study the structure of generator polynomials for Hermitian dual-containing constacyclic codes (Theorems 8 and 9), and obtain parameters of quantum codes using the Hermitian construction. The approach we used to derive Hermitian dual-containing conditions via the hull has not been used earlier. As an application, we obtain several optimal and near-to-optimal LCD codes, constacyclic codes having small hull dimensions, and quantum codes. [ABSTRACT FROM AUTHOR]