Exact maximization conjecture for minimal codewords in binary linear codes of codimension three

Let $C$ be a binary linear $[k+3,k]_2$ code, and let $M_2(k+3,k)$ denote the maximum possible number of minimal codewords among such codes. For $k\geq 4$, write

$$\mathbf{a}=\left(a_{000},a_{100},a_{010},a_{001},a_{110},a_{101},a_{011},a_{111}\right),$$

where the entries count the multiplicities of the corresponding information vectors, and let $M_2(k+3,k)$ be evaluated by the formula of Proposition~. Exact maximization conjecture. For $k\geq 4$, the exact value of $M_2(k+3,k)$ is given by that formula with $\mathbf{a}$ equal to

$$\left\{ \begin{array}{rcl} (l,l,l,l+1,l+1,l+1,l+1) &:& k=4+7l,\\ (l,l,l,l+1,l+1,l+1,l+2) &:& k=5+7l,\\ (l,l,l,l+1,l+1,l+2,l+2) &:& k=6+7l,\\ (l,l,l,l+1,l+2,l+2,l+2) &:& k=7+7l,\\ (l+1,l,l,l+2,l+2,l+1,l+2) &:& k=8+7l,\\ (l+1,l,l,l+2,l+2,l+2,l+2) &:& k=9+7l,\\ (l+1,l+1,l,l+2,l+2,l+2,l+2) &:& k=10+7l \end{array} \right.$$

if $k\leq 26$; if $k\not\equiv 0\pmod 4$ and $k\geq 27$, then

$$a_{000}=a_{001}=a_{110}=a_{111}=0,\quad a_{100}=\left\lfloor\frac{k}{4}\right\rfloor,\quad a_{010}=\left\lfloor\frac{k+1}{4}\right\rfloor,\quad a_{101}=\left\lfloor\frac{k+2}{4}\right\rfloor,\quad a_{011}=\left\lfloor\frac{k+3}{4}\right\rfloor;$$

if $k\equiv 0\pmod 4$ and $k\geq 27$, then

$$a_{000}=a_{001}=a_{110}=a_{111}=0,\quad a_{100}=\frac{k}{4},\quad a_{010}=\frac{k}{4}-1,\quad a_{101}=\frac{k}{4}+1,\quad a_{011}=\frac{k}{4}.$$

The conjecture would determine the exact extremal number of minimal codewords for binary codes with codimension three. Its large-$k$ cases are motivated by comparing the leading fourth-degree term, but the exact maximization of the underlying multiplicity formula is described as technically challenging and remains open.