🤖 AI Summary
This study addresses the lack of understanding regarding the Hermitian hull dimension of $(L,P)$-twisted generalized Reed–Solomon (TGRS) codes by focusing on a specific class of codes, denoted $C_k(\mathbf{a})$, where the defining vector $\mathbf{a}$ has length $i(q-1)$. By carefully analyzing the parity of $i$ and its relationship with $q+1$, the problem is partitioned into three distinct cases, enabling the first complete characterization of the Hermitian hull dimension for this family of TGRS codes. Leveraging this result, the authors successfully construct two new families of entanglement-assisted quantum error-correcting codes. Integrating algebraic coding theory, properties of finite fields, and the structure of the Hermitian inner product, this work not only systematically resolves the Hermitian hull dimension problem for these TGRS codes but also opens new avenues for quantum code design.
📝 Abstract
Determining the hull of linear codes has long been an important topic in coding theory. Recently, non-generalized Reed-Solomon (in short, non-GRS) codes have attracted extensive research interest. The (L,P)-twisted generalized Reed-Solomon (in short, (L,P)-TGRS) code, which is an extension of the generalized Reed-Solomon (GRS) code, constitutes a well-studied calss of non-GRS codes.There are numerous works focusing on the Euclidean hull of (L,P)-TGRS codes, while only a few results on the Hermitian hull of (L,P)-TGRS codes. In this paper, we focus on a class of (L,P)-TGRS codes C_k(a). By taking a special class of the vector a with length i(q-1), and analyze the parity of i and the relation between i and q+1, we divide three cases to fully determine the Hermitian hull dimension of C_k(a). As an application, we construct two classes of entanglement-assisted quantum error-correcting codes.