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：在其开创性1989年工作中（IEEE Trans. Inf. Theory 35(3):655-657），Roth与Lempel构造了一族著名的非Reed-Solomon型极大距离可分（MDS）码。数十年来，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持续受到研究关注。近期（2025年），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）性质显式判别准则尚未建立，且其Hermitian自正交GRL码的系统性构造亦未见报道，这限制了其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，针对GRL码两个最广泛使用的子类，给出NMDS性质的显式充要条件。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，基于所提出的Hermitian自正交GRL码，构造了四族量子GRL码，包括两族达到量子Singleton界减一的无限族量子NMDS码。与已知量子纠错码相比，我们获得了众多新型或改进的量子纠错码。本研究弥合了经典GRL码族与量子纠错应用之间的鸿沟。