Improve or verify the best-known [96,40] linear code over GF(2): current bounds 20 ≤ d ≤ 26

A linear $[n,k]_q$ code is a $k$-dimensional subspace of $\mathbb{F}_{2}^n$; its minimum distance $d$ is the smallest Hamming weight of a nonzero codeword. For $[n,k]=[96,40]$ over $\mathbb{F}_{2}$ the largest achievable minimum distance is unknown: the best-known code has $d=20$ and the best nonexistence (upper) bound is $26$, i.e. $20 \le d_{\max}(96,40) \le 26$. The lower bound is met by an explicit best-known code (a subcode of a shortened quadratic-residue code); the upper bound is a nonexistence bound. Question: construct a linear $[96,40]_{2}$ code with minimum distance $\ge 21$ (a new record, improving the lower bound), or independently verify that the recorded best-known $[96,40,20]_{2}$ code has minimum distance exactly $20$.

Acceptance. FULLY RESOLVES (improve): a generator matrix $G\in\mathbb{F}_{2}^{40\times96}$ with $\mathrm{rank}(G)=40$ whose minimum distance $d'\ge 21$, certified by a minimum-distance computation (e.g. Brouwer–Zimmermann in Magma/GAP/SageMath); $d'=26$ would settle the cell. PARTIAL: independently recompute, with a re-runnable script, the minimum distance of the recorded best-known $[96,40,20]_{2}$ code from its generator matrix and confirm it equals $20$ (reproduction of the record). Provide $G$ and the verification code.