The Hermitian Hull Dimensions for a Class of (L, P)-Twisted Generalized Reed-Solomon Codes

Jul 16, 2026

Upload by Ivy

Determining the hull of linear codes is central in coding theory, and the Hermitian hull dimension has wide algorithmic and construction significance. This paper studies a specific family of (L, P)-twisted generalized Reed–Solomon (TGRS) codes Ck(α). Using a structured choice of the vector α and analyzing the parity of an index i together with the relationship between i and q+1, the work splits the analysis into three cases to fully determine the Hermitian hull dimension of Ck(α). As an application, it constructs entanglement-assisted quantum error-correcting codes.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages35

Reading Time1 h 28 min

Introduction

Hull of linear codes and Hermitian dual

Background on non-GRS and twisted generalized Reed–Solomon codes

Related work on Hermitian hulls

Main focus and code family studied

Special choice of vector α and case analysis

Application to entanglement-assisted quantum codes

1 arXiv :2607 .03802v 1 [ cs .IT] 4 Jul 2026 The Hermitian Hull Dimensions for a Class of (L, P)-Twisted Generalized Reed–Solomon Codes Chenlu Jia, Zhonghao Liang, Yue Huang and Qunying Liao 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 Ck (α) . By taking a special class of the vector α 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 Ck (α) . As an application, we construct two classes of entanglement-assisted quantum error-correcting codes. Index Terms (L, P)-TGRS code; Hermitian hull; Entanglement-assisted quantum code. I. INTRODUCTION Let Fq be the finite field with q elements, where q is a prime power. An [n, k, d]q2 linear code C is a k-dimensional subspace of Fnq2 with minimum distance d. The Hermitian inner product of two vectors a = (a1 ,..., an) and b = (b1 ,..., bn) over Fnq2 n is defined by ⟨a, b⟩H = P aibqi . The Hermitian dual of C is i=1 C ⊥H = {x ∈ Fnq2 : ⟨x, c⟩H = 0 , for all c ∈ C} . For a linear code C , the hull Hull(C) is defined as the intersection of C and its dual code. It is well-known that the value of dim(Hull(C)) plays a critical role in determining the computational complexity of algorithms to check the permutation equivalence of two linear codes[38], computing the automorphism group of a linear code[21], calculating the number of shared pairs that required to construct an entanglement-assisted quantum error-correcting code (in short, EAQECC)[13] . And so, it is very important to determine dim(Hull(C))[4, 5, 9, 10, 12, 31, 33] . In recent years, the construction of non-GRS type linear codes has attracted considerable attention due to that they can effectively resist the Sidelnikov-Shestakov attack and the Wi-eschebrink attack. So far, there are extensive study on the properties and constructions of non-GRS codes[1, 2, 22–25, 29, 30, 41, 44, 50, 51] . In particular, in 2017, Beelen et al. [3] firstly introduced the twisted generalized Reed-Solomon (in short, TGRS) code. Subsequently, many scholars studied the TGRS code, includingthe NMDS properties, self-dual properties, self-orthogonal properties, and so on[8, 11, 15, 17, 19, 34, 40, 47, 52, 53] . In 2025, Zhao et al.[48] generalized the definition of the TGRS code to be the arbitrary twisted generalized Reed-Solomon (in short, A-TGRS) code. And then they constructed several classes of Hermitian self-dual A-TGRS codes[49] . Recently, Hu et al.[16] generalized TGRS codes to be the most general form, namely, (L, P)-twisted generalized Reed-Solomon (in short,(L, P)-TGRS) codes, and presented an in-depth and comprehensive investigation. So far, there are many study focusing on some special (L, P)-TGRS codes[14, 16, 26, 27, 36, 46] . To date, there are numerous works focused on Euclidean hulls of (L, P)-TGRS codes[7, 12, 17, 18, 28, 35, 39, 40, 42, 43, 45] . However, there exist only a few results on Hermitian hulls of (L, P)-TGRS codes, as listed below. • In 2021, Wu et al.[43] constructively proved that there exist (L, P)-TGRS codes C (L, P , B 1 ) with zero-dimensional Hermitian hull, where B 1 = 􀀰 0h−01×t−11 􀁀 0k−h×t−1 0h−1× 1 dh,t 0k−h× 1 0h−1×n−k−t􀀱 00k×h−n−kt 􀁁 k×(n−k) (0 ≤ h ≤ k − 1 , 0 ≤ t ≤ n − k − 1) . Corresponding [author: Qunying Liao. Emails:3120193984@qq.com](author: Qunying Liao. Emails:3120193984@qq.com); [liangzhongh0807@163.com](liangzh

FAQs

What is the main problem addressed in the paper?

The paper determines the Hermitian hull dimension for a class of (L, P)-twisted generalized Reed–Solomon codes Ck(α).

How does the paper approach the Hermitian hull dimension computation?

It uses a special structured vector α, then analyzes three cases based on the parity of an index i and the relationship between i and q+1.

What application does the paper provide beyond classical coding theory?

It uses the resulting Hermitian hull information to construct entanglement-assisted quantum error-correcting codes.

Jul 16, 2026

Upload by Ivy