  In this work, we study linear error-correcting codes against adversarial
insertion-deletion (insdel) errors, a topic that has recently gained a lot of
attention. We construct linear codes over $\mathbb{F}_q$, for
$q=\text{poly}(1/\varepsilon)$, that can efficiently decode from a $\delta$
fraction of insdel errors and have rate $(1-4\delta)/8-\varepsilon$. We also
show that by allowing codes over $\mathbb{F}_{q^2}$ that are linear over
$\mathbb{F}_q$, we can improve the rate to $(1-\delta)/4-\varepsilon$ while not
sacrificing efficiency. Using this latter result, we construct fully linear
codes over $\mathbb{F}_2$ that can efficiently correct up to $\delta < 1/54$
fraction of deletions and have rate $R = (1-54\cdot \delta)/1216$. Cheng,
Guruswami, Haeupler, and Li [CGHL21] constructed codes with (extremely small)
rates bounded away from zero that can correct up to a $\delta < 1/400$ fraction
of insdel errors. They also posed the problem of constructing linear codes that
get close to the half-Singleton bound (proved in [CGHL21]) over small fields.
Thus, our results significantly improve their construction and get much closer
to the bound.
