AI 中文总结
该研究分析二元单奇偶校验乘积码的结构与枚举性质，推导其权重分布的闭式表达式，提出无需穷举码字的精确计算方法，为编码理论提供了关键理论结果与计算方案。
AI 中文摘要
我们研究二元单奇偶校验乘积码的结构与枚举性质。对每个$n\geq2$，$\operatorname{SPC}(n)$表示长度为$n$的二元单奇偶校验码，由所有汉明重量为偶数的长度$n$二元向量构成。我们确定乘积码$\mathcal{C}_{m,n}=\operatorname{SPC}(m)\otimes\operatorname{SPC}(n)$的广义汉明重量层级，其码字可表示为每行每列汉明重量均为偶数的$m\times n$二元矩阵。对于平方乘积$\mathcal{C}_n=\operatorname{SPC}(n)\otimes\operatorname{SPC}(n)$，我们还确定了最大码字重量，并证明其齐次重量枚举式对称当且仅当$n$为偶数。在刻画对偶码后，我们将沃尔什-阿达马形式的麦克威廉斯恒等式应用于推导重量枚举式的精确闭式表达式。通过按汉明重量对辅助二元向量分组，我们获得了每个系数的显式公式，该公式涉及二项式系数和交替卷积。最后，利用克拉夫丘克（Krawtchouk）多项式，我们提出了一种无需穷举所有码字即可计算完整权重分布的精确方法，数值示例验证了这些公式及计算结果。
英文摘要
We investigate structural and enumerative properties of binary single parity-check product codes. For each $n\geq 2$, $\operatorname{SPC}(n)$ denotes the binary single parity-check code of length $n$, consisting of all binary vectors of length $n$ having even Hamming weight. We determine the generalized Hamming weight hierarchy of the product code $\mathcal{C}_{m,n}=\operatorname{SPC}(m)\otimes\operatorname{SPC}(n)$, whose codewords can be represented as $m\times n$ binary matrices in which every row and every column has even Hamming weight. For the square product $\mathcal{C}_n =\operatorname{SPC}(n)\otimes\operatorname{SPC}(n)$, we also determine the maximum codeword weight and prove that its homogeneous weight enumerator is symmetric if and only if $n$ is even. After characterizing the dual code, we apply the MacWilliams identity in its Walsh--Hadamard formulation to derive an exact closed-form expression for the weight enumerator. By grouping the auxiliary binary vectors according to their Hamming weights, we obtain an explicit formula for each coefficient in terms of binomial coefficients and alternating convolutions. Finally, using Krawtchouk polynomials, we present an exact procedure for computing the full weight distribution without exhaustively enumerating all codewords. Numerical examples illustrate the formulas and verify the resulting computations.
