Explicit and Efficient Constructions of Linear Codes Against Adversarial Insertions and Deletions

Abstract

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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {F}_{q}$ </tex-math></inline-formula> , for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q= {\mathrm {poly}}(1/\varepsilon)$ </tex-math></inline-formula> , that can efficiently decode from a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula> fraction of insdel errors and have rate <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(1-4\delta)/8-\varepsilon $ </tex-math></inline-formula> . We also show that by allowing codes over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {F}_{q^{2}}$ </tex-math></inline-formula> that are linear over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {F}_{q}$ </tex-math></inline-formula> , we can improve the rate to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(1-\delta)/4-\varepsilon $ </tex-math></inline-formula> while not sacrificing efficiency. Using this latter result, we construct fully linear codes over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {F}_{2}$ </tex-math></inline-formula> that can efficiently correct up to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\delta &lt; 1/54$ </tex-math></inline-formula> fraction of deletions and have rate <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$R = (1-54\cdot \delta)/1216$ </tex-math></inline-formula> . Cheng <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> (2021) constructed codes with (extremely small) rates bounded away from zero that can correct up to a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\delta &lt; 1/400$ </tex-math></inline-formula> fraction of insdel errors. They also posed the problem of constructing linear codes that get close to the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">half-Singleton bound</i> [proved in Cheng <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> (2021)] over small fields. Thus, our results significantly improve their construction and get much closer to the bound.