线性代码重量范围 (On weight spectrum of linear codes)

We study sequences of linear or affine codes with uniform weight spectrum, i.e., a part of codewords with any fixed weight tends to zero. It is proved that a sequence of linear codes has a uniform weight spectrum if the number of vectors from codes with weight $1$ grows to infinity. We find an example of a sequence of linear codes such that the dimension of the code is the half of the codelength but it has not a uniform weight spectrum. This example generates eigenfunctions of the Fourier transform with minimal support and partial covering sets. Moreover, we generalize some MacWilliams-type identity. Keywords: weight distribution of code, dual code, MacWilliams identity, Fourier transform, partial covering array

翻译：我们研究的是具有统一重量频谱的线性或方形代码序列,即具有任何固定重量的编码词组的一部分往往为零。如果从重量代码的矢量数量增长到无限,则线性代码序列具有统一的重量频谱。我们发现了一个线性代码序列的例子,这样代码的尺寸是代码长度的一半,但它没有统一的重量频谱。这个示例生成了Fourier变形的叶质功能,但支持和部分覆盖各组。此外,我们将一些MacWilliams型的特性概括化。关键词:代码的重量分布、双代码、MacWillims身份、Fourier变形、部分覆盖阵列。