Constructing Linear Codes with Maximum Distance Separable Hermitian Hulls

Core Concepts

This paper introduces tools to study linear codes whose Hermitian hulls are maximum distance separable (MDS), and proposes explicit constructions of such codes. The authors consider Hermitian hulls of both Reed-Solomon and non Reed-Solomon types of linear MDS codes, demonstrating that the codes from their constructions have larger dimensions compared to those in the literature.

Abstract

The paper focuses on studying linear codes whose Hermitian hulls are MDS. It develops several key techniques and results:

1. Lemma 8 presents necessary and sufficient conditions for a generalized Reed-Solomon (GRS) code to have a Hermitian hull containing another GRS code. This involves examining the punctured code of the given GRS code.

2. Theorem 9 simplifies the conditions in Lemma 8 and provides a sufficient condition for a GRS code to have an MDS Hermitian hull. This is used to construct new classes of Hermitian hulls in Theorems 10 to 13.

3. The paper generalizes the study of Hermitian hulls beyond Reed-Solomon-type codes. It explicitly determines the Hermitian hull dimension of a special class of two-point rational algebraic geometry (AG) codes in Theorem 18, and extends the result in Theorem 20.

4. Corollaries 1 to 3 provide more explicit constructions of families of linear codes whose Hermitian hulls are MDS. These codes are non Reed-Solomon-type and not monomially equivalent to known Hermitian self-orthogonal codes.

5. Theorem 19 shows how to enlarge the dimensions of the Hermitian hulls of some known GRS codes, leading to new parameters for entanglement-assisted quantum error-correcting codes (EAQECCs) that can improve the error-control capability compared to known stabilizer codes.

Stats

Let q be a prime power and Fq be the finite field with q elements. An [n, k, d]q2 linear code C is a k-dimensional subspace of Fnq2 with minimum (Hamming) distance d. Such a code is maximum distance separable (MDS) if d = n - k + 1.
The Hermitian inner product of vectors u = (u1, ..., un), v = (v1, ..., vn) ∈ Fnq2 is ⟨u, v⟩H = Σni=1 uivqi.
The Hermitian dual of C is C⊥H = {u ∈ Fnq2 : ⟨u, v⟩H = 0 for all v ∈ C}. The Hermitian hull of C is HullH(C) = C ∩ C⊥H.

Quotes

"Hermitian hulls of linear codes are interesting for theoretical and practical reasons alike. In terms of recent application, linear codes whose hulls meet certain conditions have been utilized as ingredients to construct entanglement-assisted quantum error correcting codes."
"Characterizing Hermitian hulls which themselves are MDS appears to be more involved and has not been extensively studied."

Key Insights Distilled From

On Linear Codes Whose Hermitian Hulls are MD

by Gaojun Luo,L... at arxiv.org 04-09-2024

https://arxiv.org/pdf/2404.04993.pdf

Deeper Inquiries

How can the techniques developed in this paper be extended to study Hermitian hulls of other classes of linear codes beyond GRS and AG codes

The techniques developed in the paper for studying Hermitian hulls of GRS and AG codes can be extended to other classes of linear codes by considering different constructions and properties specific to those codes. One approach could be to explore the Hermitian hulls of cyclic codes, BCH codes, or LDPC codes. For cyclic codes, the structure of the generator matrix and the properties of the dual code can be leveraged to analyze the Hermitian hull. In the case of BCH codes, the algebraic properties of the code can be used to determine the dimensions and properties of the Hermitian hull. For LDPC codes, the sparse parity-check matrix can provide insights into the structure of the Hermitian hull and its relationship to the original code. By adapting the tools and techniques developed for GRS and AG codes to these different classes of linear codes, a comprehensive understanding of Hermitian hulls across various types of codes can be achieved.

What are the potential applications of linear codes with MDS Hermitian hulls beyond quantum error correction

Linear codes with MDS Hermitian hulls have applications beyond quantum error correction in various fields such as cryptography, telecommunications, and data storage. One potential application is in secure communication systems where the codes can be used for error detection and correction in data transmission. These codes can also be utilized in fault-tolerant systems to ensure reliable operation in the presence of errors. In data storage systems, linear codes with MDS Hermitian hulls can enhance the efficiency and reliability of storage mechanisms by providing robust error correction capabilities. Additionally, in distributed computing and cloud storage, these codes can play a crucial role in maintaining data integrity and security during data transfer and storage operations. Overall, the applications of linear codes with MDS Hermitian hulls extend to various domains where error detection and correction are essential for reliable and secure data processing.

Can the constructions in this paper be generalized to study Hermitian hulls of linear codes over other finite fields or rings

The constructions presented in the paper can be generalized to study Hermitian hulls of linear codes over other finite fields or rings by adapting the techniques to the specific algebraic structures and properties of those fields or rings. For linear codes over different finite fields, such as prime fields or extension fields, the properties of the field elements and the structure of the codes would influence the construction and analysis of the Hermitian hulls. Similarly, for linear codes over rings, such as integer rings or polynomial rings, the algebraic properties of the ring elements and the code structures would need to be considered in studying the Hermitian hulls. By modifying the constructions and methodologies to suit the characteristics of the chosen finite fields or rings, a comprehensive study of Hermitian hulls of linear codes in diverse algebraic settings can be achieved.