New Quantum MDS Codes with Flexible Parameters from Hermitian Self-Orthogonal GRS Codes

TL;DR

This work introduces a flexible Hermitian self-orthogonal construction of quantum MDS codes via evaluation codes built from generalized Reed-Solomon codes. By carefully selecting parameter triples $(\lambda,\tau,\rho)$ with $\lambda|q-1$, $\tau|q+1$, $\rho|q+1$, and a twist vector chosen through an explicit $L$-dependent design, the authors identify failure points of Hermitian self-orthogonality and determine a distance-cap $T$ that yields $[[n,n-2d+2,d]]_q$ stabilizer codes for all $2\le d\le T$, with $n=\lambda\tau\sigma$ and $2\le \sigma\le \rho/\kappa$. The main theorem provides explicit cases (based on parity and auxiliary conditions) for the achievable distance bound and demonstrates that the resulting codes include many new parameter sets not covered by prior constructions, including lengths not divisible by $q-1$ or $q+1$. The paper substantiates these results with numerous examples and connections to existing parameter families, showing both novelty and compatibility with known quantum MDS code families.

Abstract

Let $q$ be a prime power. Let $λ>1$ be a divisor of $q-1$, and let $τ>1$ and $ρ>1$ be divisors of $q+1$. Under certain conditions we prove that there exists an MDS stabilizer quantum code with length $n=λτσ$ where $2\le σ\le ρ$. This is a flexible construction, which includes new MDS parameters not known before.

Theorems & Definitions (46)

• Theorem 1.1
• Proposition 2.1
• Proposition 2.2
• Theorem 2.3
• Definition 2.4
• Remark 2.5
• Lemma 3.1
• proof
• Lemma 3.2
• proof