In their seminal 1989 work (IEEE Trans. Inf. Theory 35(3):655-657), Roth and Lempel constructed a well-known family of non-Reed-Solomon maximum distance separable (MDS) codes. For decades, this family of codes has attracted extensive research attention due to its algebraic structure, low-complexity decoding, and broad applications in cryptography and data storage. Most recently, in 2025, the generalized Roth-Lempel (GRL) framework unifies Roth-Lempel codes and its extensions under a flexible algebraic structure. However, explicit criteria for the near-MDS (NMDS) property of GRL codes have not been established, and no systematic construction of Hermitian self-orthogonal GRL codes has been reported, limiting their deployment in classical and quantum error correction. In this work, we make three contributions to address these gaps. First, we give explicit necessary and sufficient conditions for the NMDS property of the two most widely used subclasses of GRL codes. Second, we construct four new families of Hermitian self-orthogonal codes from GRL codes. Two of these families are NMDS, with parameters not covered by existing Hermitian self-orthogonal NMDS codes. Third, based on the proposed Hermitian self-orthogonal GRL codes, we construct four families of quantum GRL codes, including two infinite families of quantum NMDS codes that attain the quantum Singleton bound minus one. Compared to the known quantum error-correcting codes, we obtain many new or improved quantum error-correcting codes. This work bridges the gap between classical GRL code families and quantum error-correction applications.

翻译：在Roth与Lempel1989年的开创性工作（IEEE Trans. Inf. Theory 35(3):655-657）中，他们构造了一个著名的非Reed-Solomon型极大距离可分（MDS）码族。几十年来，该码族因其代数结构、低复杂度译码以及在密码学与数据存储中的广泛应用而备受研究关注。最近，2025年提出的广义Roth-Lempel（GRL）框架通过灵活的代数结构统一了Roth-Lempel码及其扩展形式。然而，GRL码的近MDS（NMDS）性质的显式判定准则尚未建立，且尚无关于Hermitian自正交GRL码的系统性构造报道，这限制了它们在经典与量子纠错中的应用。本文针对上述空白做出三项贡献。首先，我们给出了GRL码两个最广泛使用子类具备NMDS性质的充要条件。其次，基于GRL码构造了四族新的Hermitian自正交码，其中两族为NMDS码，其参数未被现有Hermitian自正交NMDS码覆盖。第三，利用所提出的Hermitian自正交GRL码，构造了四族量子GRL码，包括两族达到量子Singleton界减一的无限量子NMDS码族。与已知量子纠错码相比，我们获得了许多新型或改进的量子纠错码。本文工作弥合了经典GRL码族与量子纠错应用之间的鸿沟。