The determination of the weight distribution of linear codes has been a fascinating problem since the very beginning of coding theory. There has been a lot of research on weight enumerators of special cases, such as self-dual codes and codes with small Singleton's defect. We propose a new set of linear relations that must be satisfied by the coefficients of the weight distribution. From these relations we are able to derive known identities (in an easier way) for interesting cases, such as extremal codes, Hermitian codes, MDS and NMDS codes. Moreover, we are able to present for the first time the weight distribution of AMDS codes. We also discuss the link between our results and the Pless equations.

中文翻译：

---

线性码权重分布公式在AMDS码中的应用

自编码理论诞生以来，确定线性码的权重分布一直是一个引人入胜的问题。对特殊情况的权重枚举器已有大量研究，例如自对偶码和具有小单例缺陷的码。我们提出了一组新的线性关系，必须满足权重分布的系数。从这些关系中，我们能够为有趣的情况（例如极值代码、厄米代码、MDS 和 NMDS 代码）推导出已知的身份（以更简单的方式）。此外，我们能够首次展示AMDS代码的权重分布。我们还讨论了我们的结果与 Pless 方程之间的联系。