Improve or verify the best-known [48,24] linear code over GF(5): current bounds 15 ≤ d ≤ 20

A linear $[n,k]_q$ code is a $k$-dimensional subspace of $\mathbb{F}_{5}^n$; its minimum distance $d$ is the smallest Hamming weight of a nonzero codeword. For $[n,k]=[48,24]$ over $\mathbb{F}_{5}$ the largest achievable minimum distance is unknown: the best-known code has $d=15$ and the best nonexistence (upper) bound is $20$, i.e. $15 \le d_{\max}(48,24) \le 20$. The lower bound is met by an explicit best-known code (codetables construction type GHa); the upper bound is a nonexistence bound. Question: construct a linear $[48,24]_{5}$ code with minimum distance $\ge 16$ (a new record, improving the lower bound), or independently verify that the recorded best-known $[48,24,15]_{5}$ code has minimum distance exactly $15$.

Acceptance. FULLY RESOLVES (improve): a generator matrix $G\in\mathbb{F}_{5}^{24\times48}$ with $\mathrm{rank}(G)=24$ whose minimum distance $d'\ge 16$, certified by a minimum-distance computation (e.g. Brouwer–Zimmermann in Magma/GAP/SageMath); $d'=20$ would settle the cell. PARTIAL: independently recompute, with a re-runnable script, the minimum distance of the recorded best-known $[48,24,15]_{5}$ code from its generator matrix and confirm it equals $15$ (reproduction of the record). Provide $G$ and the verification code.