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 Fq2, linear over Fq, with rate 1/2 − δ − ε that can efficiently correct a δ-fraction of indel errors, whereq=O(ε−4). Additionally, we construct fully linear codes over Fqwith rate 1/2 − 2 √ δ − ε that can also efficiently correct δ-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 codeC⊆ Fnlinear over E ⊂ F a subfield of F, such thatChas the ability to correct δ-fraction of indels, the rate is bounded by (1 − δ)/2.

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$q=O\left(\varepsilon^{-4}\right)$. 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.