AI 中文总结
该研究针对线性码的调和高权分布给出西蒙尼斯型MacWilliams恒等式的组合证明，推导了随机线性码调和高权枚举子的矩相关性质及协方差公式。
AI 中文摘要
我们针对线性码的调和高权分布，给出了西蒙尼斯型MacWilliams恒等式的组合证明。此外，我们研究了随机线性码的调和高权枚举子的统计矩。通过线性码生成矩阵的秩函数定义该枚举子，我们证明，由于随机矩阵的固有对称性，对于所有非平凡调和函数，其期望均为零；我们还推导出了一个明确的非平凡协方差公式。
英文摘要
We present a combinatorial proof of Simonis type MacWilliams identity for harmonic higher weight distributions of linear codes. Furthermore, we investigate the statistical moments of the harmonic higher weight enumerators for random linear codes. Defining the enumerators via rank functions of the generator matrices of linear codes, we prove that its expectation vanishes for all non-trivial harmonic functions due to the inherent symmetry of random matrices, and we also derive an explicit, non-trivial formula for the covariance.
Comments19 pages
