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Explicit and Efficient Constructions of Linear Codes Against Adversarial Insertions and Deletions
IEEE Transactions on Information Theory ( IF 2.5 ) Pub Date : 2022-05-09 , DOI: 10.1109/tit.2022.3173185
Roni Con 1 , Amir Shpilka 1 , Itzhak Tamo 2
Affiliation  

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 $q= {\mathrm {poly}}(1/\varepsilon)$ , 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 et al. (2021) 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 Cheng et al. (2021)] over small fields. Thus, our results significantly improve their construction and get much closer to the bound.

中文翻译:

针对对抗性插入和删除的线性代码的显式和有效构造

在这项工作中,我们研究了针对对抗性插入删除(insdel)错误的线性纠错码,这是一个最近引起很多关注的话题。我们构造线性码 $\mathbb {F}_{q}$ , 为了 $q= {\mathrm {poly}}(1/\varepsilon)$ ,可以有效地从 $\三角洲$insdel 错误的分数和有率 $(1-4\delta)/8-\varepsilon $ . 我们还通过允许代码 $\mathbb {F}_{q^{2}}$是线性的 $\mathbb {F}_{q}$ ,我们可以将速率提高到 $(1-\delta)/4-\varepsilon $在不牺牲效率的同时。使用后一个结果,我们在 $\mathbb {F}_{2}$可以有效地纠正 $\delta < 1/54$删除率和有率 $R = (1-54\cdot \delta)/1216$ . 程等。(2021 年)构建了具有(极小)速率的代码,其范围远离零,可以纠正高达 $\delta < 1/400$insdel 错误的一小部分。他们还提出了构建接近于半单例界[在 Cheng 中证明等。(2021)]在小领域。因此,我们的结果显着改善了它们的构造并更接近界限。
更新日期:2022-05-09
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