In this paper, we have introduced the concepts of support distribution and the support enumerator as refinements of the classical weight distribution and weight enumerator respectively, capturing coordinate level activity in linear block codes. More precisely, we have established formula for counting codewords in the linear code C whose i-th coordinate is nonzero. Moreover, we derived a MacWilliam's type identity, relating the normalized support enumerators of a linear code and its dual, explaining how coordinate information transforms under duality. Using this identity we deduce a condition for self duality based on the equality of support distributions. These results provide a more detailed understanding of code structure and complement classical weight based duality theory.

翻译：本文引入了支撑分布与支撑枚举器的概念，分别作为经典重量分布与重量枚举器的细化，以刻画线性分组码中坐标层面的活跃性。具体而言，我们建立了计算线性码 C 中第 i 个坐标非零的码字数量的公式。此外，我们推导了 MacWilliams 型恒等式，该等式关联了一个线性码与其对偶码的归一化支撑枚举器，从而解释了坐标信息在对偶关系下的变换规律。利用这一恒等式，我们基于支撑分布的相等性推导了自对偶码的一个判定条件。这些结果为理解码的结构提供了更细致的视角，并补充了经典的基于重量的对偶理论。