In this paper, we characterize the average Hamming weight distribution of subsequences of maximum-length sequences ($m$-sequences). In particular, we consider all possible $m$-sequences of dimension $k$ and find the average number of subsequences of length $n$ that have a Hamming weight $t$. To do so, we first characterize the Hamming weight distribution of the average dual code and use the MacWilliams identity to find the average Hamming weight distribution of subsequences of $m$-sequences. We further find a lower bound on the minimum Hamming weight of the subsequences and show that there always exists a primitive polynomial to generate an $m$-sequence to meet this bound. We show via simulations that when a proper primitive polynomial is chosen, subsequences of the $m$-sequence can form a good rateless code that can meet the normal approximation benchmark.

翻译：在本文中,我们给出了最大长度序列次序列平均含重重量分布的特征。 特别是, 我们考虑了所有可能的维度以百万美元计重, 并找到了长度以美元计值的次序列的平均数量。 为此, 我们首先给出了平均双重代码的含重重量分布的特征, 并使用 MacWilliams 身份来找到最大序列次序列以美元计值的平均含重分布。 我们进一步发现后继序列最低含重的下限, 并显示始终存在一个原始的多元序列来生成一个美元序列来达到这一界限。 我们通过模拟来显示, 当选择适当的原始多元值时, 美元后继的子序列可以形成一个符合正常近似基准的好的无息值代码 。