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On Linear Codes Whose Hermitian Hulls are MD

Gaojun Luo, Lin Sok, Martianus Frederic Ezerman, San Ling

TL;DR

The paper addresses the problem of constructing linear codes over ${\mathbb F}_{q^2}$ whose Hermitian hulls are maximum-distance-separable (MDS) and analyzes their implications for entanglement-assisted quantum error-correcting codes (EAQECCs). It develops tools to study Hermitian hulls and delivers explicit constructions from Generalized Reed-Solomon (GRS) codes and from two-point rational algebraic-geometry (AG) codes, achieving hulls that are MDS and, for fixed hull-dimension, codes with larger overall dimensions than prior work. The authors show that these hull-structured codes enable MDS EAQECCs with improved minimum distances, by leveraging hull dimensions to trade entanglement resources for distance. This yields a broader, more-flexible design space for quantum error correction, including non-Reed-Solomon type codes that are not monomially equivalent to RS-type codes, with practical implications for qudit-based quantum communication.

Abstract

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. This family of quantum codes is often seen as a generalization of quantum stabilizer codes. Theoretically, compared with the Euclidean setup, the Hermitian case is much harder to deal with. Hermitian hulls of MDS linear codes with low dimensions have been explored, mostly from generalized Reed-Solomon codes. Characterizing Hermitian hulls which themselves are MDS appears to be more involved and has not been extensively studied. This paper introduces some tools to study linear codes whose Hermitian hulls are MDS. Using the tools, we then propose explicit constructions of such codes. We consider Hermitian hulls of both Reed-Solomon and non Reed-Solomon types of linear MDS codes. We demonstrate that, given the same Hermitian hull dimensions, the codes from our constructions have dimensions which are larger than those in the literature.

On Linear Codes Whose Hermitian Hulls are MD

TL;DR

The paper addresses the problem of constructing linear codes over whose Hermitian hulls are maximum-distance-separable (MDS) and analyzes their implications for entanglement-assisted quantum error-correcting codes (EAQECCs). It develops tools to study Hermitian hulls and delivers explicit constructions from Generalized Reed-Solomon (GRS) codes and from two-point rational algebraic-geometry (AG) codes, achieving hulls that are MDS and, for fixed hull-dimension, codes with larger overall dimensions than prior work. The authors show that these hull-structured codes enable MDS EAQECCs with improved minimum distances, by leveraging hull dimensions to trade entanglement resources for distance. This yields a broader, more-flexible design space for quantum error correction, including non-Reed-Solomon type codes that are not monomially equivalent to RS-type codes, with practical implications for qudit-based quantum communication.

Abstract

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. This family of quantum codes is often seen as a generalization of quantum stabilizer codes. Theoretically, compared with the Euclidean setup, the Hermitian case is much harder to deal with. Hermitian hulls of MDS linear codes with low dimensions have been explored, mostly from generalized Reed-Solomon codes. Characterizing Hermitian hulls which themselves are MDS appears to be more involved and has not been extensively studied. This paper introduces some tools to study linear codes whose Hermitian hulls are MDS. Using the tools, we then propose explicit constructions of such codes. We consider Hermitian hulls of both Reed-Solomon and non Reed-Solomon types of linear MDS codes. We demonstrate that, given the same Hermitian hull dimensions, the codes from our constructions have dimensions which are larger than those in the literature.
Paper Structure (12 sections, 26 theorems, 79 equations, 3 tables)

This paper contains 12 sections, 26 theorems, 79 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Known Results on Hulls
  3. Hermitian Hulls which are MDS
  4. Our Contributions
  5. Preliminaries
  6. Cyclic Codes and Generalized Reed-Solomon codes
  7. Algebraic Geometry Codes
  8. Main results
  9. MDS Hermitian Hulls from GRS Codes
  10. Hermitian MDS Hulls from Two-Point Rational AG Codes
  11. Quantum error correction
  12. Concluding remarks

Key Result

Lemma 1

(Delsarte1975) Let ${\mathcal{C}}$ be a cyclic code of length $n$ over ${\mathbb F}_q$. Let $r={\rm ord}_n(q)$ and let $\alpha$ be a primitive $n^{\rm th}$ root of unity in ${\mathbb F}_{q^r}$. Let the generating set of ${\mathcal{C}}$ be $\bigcup_{\ell=1}^sC_{i_\ell}$, where $C_{i_\ell}$ is the $q$

Theorems & Definitions (50)

  • Lemma 1
  • Lemma 2
  • Proposition 3
  • Proposition 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • proof
  • Theorem 9
  • ...and 40 more