(+)-(L, P)-TGRS 码 (The (+)-(L, P)-TGRS code)

The construction of the non-Reed-Solomon (in short, non-RS) type linear code has been one of the research hotspots in recent years. In 2025, Hu et al. constructed some non-RS MDS codes by defining the (L, P)-twisted generalized Reed-Solomon code (in short, (L, P)-TGRS). In this paper, we focus on the (+)-(L, P)-TGRS code C. We firstly present a parity-check matrix. Secondly, we give a sufficient and necessary condition for C to be NMDS which partially answers two open problems proposed by Hu et al. in 2025, and prove that C is non-RS for 2k > n which partially improves the corresponding result given by Hu et al. in 2025,. Thirdly, we give a sufficient condition for C not to be self-dual or self-orthogonal, respectively, furthermore, we construct two classes of self-orthogonal codes which is a promotion of the corresponding result given by Ding et al. in 2025. Finally, some examples are given.

翻译：非里德-所罗门（简称非RS）型线性码的构造是近年来的研究热点之一。2025年，Hu等人通过定义(L, P)-扭曲广义里德-所罗门码（简称(L, P)-TGRS）构造了一些非RS MDS码。本文聚焦于(+)-(L, P)-TGRS码C。首先，我们给出了该码的校验矩阵。其次，我们给出了C成为NMDS码的充要条件，这部分回答了Hu等人在2025年提出的两个公开问题；并证明了当2k > n时C是非RS码，这部分改进了Hu等人在2025年给出的相应结果。第三，我们分别给出了C不是自对偶码或自正交码的充分条件；此外，我们构造了两类自正交码，这推广了Ding等人在2025年给出的相应结果。最后，我们给出了一些实例。