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.
翻译:在这项工作中,我们研究了针对对抗性插入-删除(insdel)错误(insdel)的线性错误校正代码,这是一个最近引起极大关注的议题。我们为$qätext{poly}(1/\varepsilon) $qthbb{F}qqq$,在$mathbb{F ⁇ qqq$($qäqqq$) 上建了线性代码,这可以有效地解码出美元(delta) 部分的内差(1-4\delta) /8\ varepsilon) 。我们还表明,如果允许代码超过$\mathb{F ⁇ 2}(美元),那么,如果允许代码线性超过$\mathb{Fäqq}$(美元),那么我们就可以在不牺牲效率的情况下将费率提高到$(1-delusb{delepsilon) 的线性代码(GHeqral-d) 上建得更高(GHY-c_roup rouple) roup roup a roup roup roup roup roup roup roup $.(GH) (Gy) roup roup rolate) (GUlate) rolate a roup d) roup d) roup roupd) roupd) roupal_ roupd) roup a_ roupd_ roupd_ roupd_ 至 1-xxxxxxxxxxxxxxxxxxxx 至 至 至 。