We determine the weight spectra of the Reed-Muller codes $RM(m-3,m)$ for $m\ge 6$ and $RM(m-4,m)$ for $m\ge 8$. The technique used is induction on $m$, using that the sum of two weights in $RM(r-1,m-1)$ is a weight in $RM(r,m)$, and using the characterization by Kasami and Tokura of the weights in $RM(r,m)$ that lie between its minimum distance $2^{m-r}$ and the double of this minimum distance. We also derive the weights of $RM(3,8),\,RM(4,9),$ by the same technique. We conclude with a conjecture on the weights of $RM(m-c,m)$, where $c$ is fixed and $m$ is large enough.

翻译：我们确定了对于$m\ge 6$的$RM(m-3,m)$码和对于$m\ge 8$的$RM(m-4,m)$码的重量谱。所采用的技术是对$m$进行归纳，利用$RM(r-1,m-1)$中两个重量之和是$RM(r,m)$中的一个重量，并借助Kasami和Tokura对$RM(r,m)$中位于其最小距离$2^{m-r}$与该最小距离两倍之间的重量的刻画。我们同样通过该技术推导出$RM(3,8)$和$RM(4,9)$的重量。最后，我们提出一个关于$RM(m-c,m)$（其中$c$固定且$m$足够大）重量的猜想。