Abstract
This paper presents a tutorial survey on error-correcting codes for deletion and insertion errors, covering both classical and quantum coding theory. We first review fundamental properties: the sizes of deletion and insertion balls, the Levenshtein (edit) distance, the equivalence between deletion-correcting and insertion-correcting codes, and upper and lower bounds on redundancy. We then discuss single deletion-correcting codes, focusing on VT (Varshamov-Tenengol'ts) codes, which achieve the optimal redundancy of log2n+O(1) with linear-time decoding algorithms for both deletion and insertion errors, along with Monotone codes and Tenengol'ts codes for extended settings. Multi-deletion codes are surveyed next, covering Helberg codes, two-deletion-correcting constructions, the efficient codes of Brakensiek–Guruswami–Zbarsky, and synchronization strings by Haeupler–Shahrasbi that approach the Singleton bound. We also discuss extensions of deletion-error research, including BAD error codes, two-dimensional array codes, and formal verification using a proof assistant. Finally, we introduce quantum deletion error-correcting codes, formulated via partial traces. We present the NH conditions—a set of combinatorial conditions on two classical sets A and B—that provide a sufficient condition for single quantum deletion correction, the minimum-length four-qubit code (n=4), permutation-invariant quantum codes for multiple deletions, and constructions derived from quantum Reed–Solomon codes.
