  Using integer linear programming and table-lookups we prove that there is no
binary linear $[1988, 12, 992]$ code. As a by-product, the non-existence of
binary linear codes with the parameters $[324, 10, 160]$, $[356, 10, 176]$,
$[772,11,384]$, and $[836,11,416]$ is shown.
  Our work is motivated by the recent construction of the extended dualized
Kerdock code $\hat{\mathcal{K}}^*_{6}$, which is a $\mathbb{Z}_4$-linear code
having a non-linear binary Gray image with the parameters $(1988,2^{12},992)$.
By our result, the code $\hat{\mathcal{K}}^*_{6}$ can be added to the small
list of $\mathbb{Z}_4$-codes for which it is known that the Gray image is
better than any binary linear code.
