Improved Constructions of Linear Codes for Insertions and Deletions

In this work, we study linear error-correcting codes against adversarial insertion-deletion (indel) errors. While most constructions for the indel model are nonlinear, linear codes offer compact representations, efficient encoding, and decoding algorithms, making them highly desirable. A key challenge in this area is achieving rates close to the half-Singleton bound for efficient linear codes over finite fields. We improve upon previous results by constructing explicit codes over $\mathbb{F}_{q^2}$, linear over $\mathbb{F}_q$, with rate $1/2 - \delta - \varepsilon$ that can efficiently correct a $\delta$-fraction of indel errors, where $q = O(\varepsilon^{-4})$. Additionally, we construct fully linear codes over $\mathbb{F}_q$ with rate $1/2 - 2\sqrt{\delta} - \varepsilon$ that can also efficiently correct $\delta$-fraction of indels. These results significantly advance the study of linear codes for the indel model, bringing them closer to the theoretical half-Singleton bound. We also generalize the half-Singleton bound, for every code $C \subseteq \mathbb{F}^n$ linear over $\mathbb{E} \subset \mathbb{F}$ a subfield of $\mathbb{F}$, such that $C$ has the ability to correct $\delta$-fraction of indels, the rate is bounded by $(1-\delta)/2$.