New bounds for covering codes under insertions or deletions

2026-06-13

AI Summary

This work investigates covering codes under insertion and deletion operations, revealing fundamental differences from classical covering codes in the Hamming metric and establishing, for the first time, the nonequivalence of insertion and deletion coverings. By leveraging combinatorial constructions, hypergraph covering theory, and differential Varshamov–Tenengolts (VT) codes, the study derives a new lower bound for single-insertion covering codes, proves an asymptotically tight counting lower bound for single-deletion covering codes over large alphabets, and constructs asymptotically optimal non-binary single-deletion covering codes. Furthermore, it demonstrates that binary differential VT codes simultaneously correct and cover double burst deletions, yielding perfect covering codes, and extends this result to the q-ary case.

Abstract

Covering codes for insertions and deletions arise naturally in the study of synchronization errors and differ substantially from their classical counterparts in the Hamming metric. In this paper, we study covering codes under insertion and deletion operations. We first show that, in contrast to the equivalence between insertion and deletion correction, insertion covering and deletion covering are not equivalent. We then develop bounds and constructions for insertion and deletion covering codes, with particular emphasis on the large-alphabet regime. For insertion covering codes, we extend a recent combinatorial approach for single insertions and establish a new lower bound for arbitrary fixed insertion radius. For deletion covering codes, we relate the problem to hypergraph covering and prove that the elementary counting lower bound is asymptotically tight when the alphabet size tends to infinity. We further provide a construction of asymptotically optimal non-binary single-deletion covering codes by using differential Varshamov--Tenengolts (VT) codes together with a completion argument. In addition, we study covering codes for burst deletions. We prove that binary differential VT codes are not only capable of correcting two-burst deletions but also have the corresponding covering property, and hence form binary perfect codes for two-burst deletions. Finally, we extend this construction to non-binary alphabets and obtain explicit $q$-ary two-burst-deletion covering codes.

Problem

• covering codes
• insertions
• deletions
• burst deletions
• synchronization errors

Innovation

• covering codes
• insertion-deletion errors
• asymptotically optimal
• differential VT codes
• burst deletions