Improve or verify the best-known [128,64] binary linear code: current bounds 22 ≤ d ≤ 28

A linear $[n,k]_q$ code is a $k$-dimensional subspace of $\mathbb{F}_q^n$; its minimum distance $d$ is the smallest Hamming weight of a nonzero codeword. For $[n,k]=[128,64]$ over $\mathbb{F}_2$ the largest achievable minimum distance is unknown: the best-known code has $d=22$ and the best nonexistence (upper) bound is $28$, i.e. $22 \le d_{\max}(128,64) \le 28$. The lower bound is met by an explicit best-known code (an extended BCH code, codetables construction type XBC); the upper bound comes from shortening bounds. Question: construct a linear $[128,64]_2$ code with minimum distance $\ge 23$ (a new record, improving the lower bound), or independently verify that the recorded best-known $[128,64,22]_2$ code indeed has minimum distance exactly $22$.

Acceptance. FULLY RESOLVES (improve): a generator matrix $G\in\mathbb{F}_2^{64\times128}$ of a linear code with $\mathrm{rank}(G)=64$ whose minimum distance $d'\ge 23$, certified by a minimum-distance computation (e.g. Brouwer–Zimmermann in Magma/GAP/SageMath); $d'=28$ would settle the cell. PARTIAL: independently recompute, with a re-runnable script, the minimum distance of the recorded best-known $[128,64,22]_2$ code from its generator matrix and confirm it equals $22$ (a reproduction of the record). Provide $G$ and the verification code.