Hermitian hull of constacyclic codes over a class of non-chain rings and new quantum codes. (English) Zbl 07872435
Summary: 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 \(\lambda\)-constacyclic codes over \(R_e = \mathbb{F}_{q^2} + u\mathbb{F}_{q^2} + u^2\mathbb{F}_{q^2} + \cdots + u^{e-1}\mathbb{F}_{q^2}, u^e=1\) where \(\mathbb{F}_{q^2}\) is the finite field of \(q^2\) elements, \(e\mid (q+1)\) and \(\lambda = \eta_1\alpha_1 + \eta_2\alpha_2 + \cdots + \eta_e\alpha_e\) for \(\alpha_l\in\mathbb{F}_{q^2}^\ast\) of order \(r_l\) such that \(r_l\mid q+1\) (for each \(1 \leq l \leq e\)). 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.
MSC:
| 94B05 | Linear codes (general theory) |
| 94B15 | Cyclic codes |
| 94B35 | Decoding |
| 94B60 | Other types of codes |
Software:
MagmaReferences:
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