Carlet–Lou weight-spectrum conjecture for Reed–Muller codes

Let $RM(r,m)$ denote the Reed–Muller code of order $r$ in $m$ variables, and let its weight spectrum be the set of Hamming weights of its codewords. For a set $S$ of weights, write $\overline{S}=\{2^m-s\mid s\in S\}$. Let $c$ be any positive integer and suppose that $m\geq 2c$. Carlet–Lou weight-spectrum conjecture. The weight spectrum of $RM(m-c,m)$ has the form

$$\{0\}\cup A\cup B\cup C\cup \overline{B}\cup \overline{A}\cup \{2^m\},$$

where $A\subseteq[2^c,2^{c+1}]$ is the set given by Kasami and Tokura, $B\subseteq[2^{c+1},2^{c+1}+2^{c-1}]$ is the set given by Kasami, Tokura, and Azumi, and $C\subseteq[2^{c+1}+2^{c-1},2^m-2^{c+1}-2^{c-1}]$ consists of all consecutive even integers. The sets $\overline{A}$ and $\overline{B}$ are the complements of $A$ and $B$, respectively, with respect to $2^m$. This conjecture proposes a general description of the weight spectrum of high-order Reed–Muller codes; the cited descriptions of $A$ and $B$ come from earlier work, while the asserted complete form remains an open question.