🤖 AI Summary
本文研究了二进制单奇偶校验积码的结构和计数特性，通过特征和方法确定了其权重分布，解决了无需枚举所有码字即可计算全权重分布的问题。
📝 Abstract
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.