Improve or verify the best-known [44,22] linear code over GF(4): current bounds 14 ≤ d ≤ 16

A linear $[n,k]_q$ code is a $k$-dimensional subspace of $\mathbb{F}_{4}^n$; its minimum distance $d$ is the smallest Hamming weight of a nonzero codeword. For $[n,k]=[44,22]$ over $\mathbb{F}_{4}$ the largest achievable minimum distance is unknown: the best-known code has $d=14$ and the best nonexistence (upper) bound is $16$, i.e. $14 \le d_{\max}(44,22) \le 16$. The lower bound is met by an explicit best-known code (a quadratic-residue (QR) code); the upper bound is a nonexistence bound. Question: construct a linear $[44,22]_{4}$ code with minimum distance $\ge 15$ (a new record, improving the lower bound), or independently verify that the recorded best-known $[44,22,14]_{4}$ code has minimum distance exactly $14$.

Acceptance. FULLY RESOLVES (improve): a generator matrix $G\in\mathbb{F}_{4}^{22\times44}$ with $\mathrm{rank}(G)=22$ whose minimum distance $d'\ge 15$, certified by a minimum-distance computation (e.g. Brouwer–Zimmermann in Magma/GAP/SageMath); $d'=16$ would settle the cell. PARTIAL: independently recompute, with a re-runnable script, the minimum distance of the recorded best-known $[44,22,14]_{4}$ code from its generator matrix and confirm it equals $14$ (reproduction of the record). Provide $G$ and the verification code.