In this paper, we study a class of special linear codes involving their parameters, weight distributions, self-orthogonal properties, deep holes, and the existence of error-correcting pairs. We prove that such codes must be maximum distance separable (MDS) codes or near MDS codes and completely determine their weight distributions with the help of the solutions to some subset sum problems. Based on the Schur method, we show that such codes are not equivalent to generalized Reed-Solomon (GRS) codes. A sufficient and necessary condition for such codes to be self-orthogonal is also characterized. Based on this condition, we further deduce that there are no self-dual codes in this class of linear codes and explicitly construct two classes of almost self-dual codes. Additionally, we find a class of deep holes of such codes and determine the existence of their error-correcting pairs in most cases, which also reveal more connections between such codes and GRS codes.

翻译：本文研究一类特殊线性码，涉及其参数、重量分布、自正交性、深洞及纠错对的存在性。我们证明此类码必为极大距离可分（MDS）码或近极大距离可分（NMDS）码，并借助若干子集和问题的解完整确定了其重量分布。基于Schur方法，我们证明此类码不等价于广义Reed-Solomon（GRS）码。同时刻画了此类码具有自正交性的充要条件。基于该条件，我们进一步推导出该类线性码中不存在自对偶码，并显式构造了两类几乎自对偶码。此外，我们发现了此类码的一类深洞，并在大多数情况下确定了其纠错对的存在性，这也揭示了此类码与GRS码之间更深层的联系。