Function-Correcting Codes for Insertion-Deletion Channel
Anamika Singh, Abhay Kumar Singh
TL;DR
This work advances the theory of computing function values over insertion–deletion (insdel) channels by introducing function-correcting codes for insdel errors (FCIDCs) and tying optimal redundancy to irregular insdel-distance codes. It defines equivalent FCC formulations (deletion, insertion, insdel) and develops irregular insdel-distance codes with distance matrices, deriving Gilbert–Varshamov and Plotkin-type bounds and simplified redundancy bounds. The authors instantiate FCIDCs for several function classes—VT-syndrome, number-of-runs, maximum-run-length, and locally bounded functions—providing bounds and concrete constructions. The framework enables significant redundancy reductions for function computation in synchronization-noise environments, with practical implications for DNA data storage and related applications where insertions and deletions dominate error models.
Abstract
In coding theory, handling errors that occur when symbols are inserted or deleted from a transmitted message is a long-standing challenge. Optimising redundancy for insertion and deletion channels remains a key open problem with significant importance for applications in DNA data storage and document exchange. Recently, a coding framework known as function-correcting codes has been proposed to address the challenge of minimising redundancy while preserving specific functions of the message. This framework has gained attention due to its potential applications in machine learning systems and long-term archival data storage. Motivated by the problem of redundancy optimisation for insertion and deletion channels, we propose a new framework called function-correcting codes for insdel channels. In this paper, we introduce the notions of function-correcting insertion codes, function-correcting deletion codes, and function-correcting insdel codes, and we show that these three formulations are equivalent. We then define insdel distance matrices and irregular insdel-distance codes, and derive lower and upper bounds on the optimal redundancy achievable by function-correcting codes for insdel channels. In addition, we establish Gilbert-Varshamov and Plotkin-like bounds on the length of irregular insdel-distance codes. Using the relation between optimal redundancy and the length of such codes, we obtain a simplified lower bound on optimal redundancy. Finally, we derive bounds on the optimal redundancy of function-correcting insdel codes for several classes of functions, including locally bounded functions, VT syndrome functions, the number-of-runs function, and the maximum-run-length function.