Single-Deletion Single-Substitution Correcting Codes

TL;DR

This paper establishes bounds and constructs codes capable of correcting a single insertion or deletion along with a single substitution, advancing error correction in DNA storage and communication systems.

Contribution

It derives a non-asymptotic upper bound on code size and provides an explicit construction with low redundancy for single-deletion single-substitution correction.

Findings

Redundancy must be at least 2 log n for such codes.

Constructed codes have at most 6 log n + 8 redundancy bits.

Compared to previous 3-deletion correction codes, this is more efficient.

Abstract

Correcting insertions/deletions as well as substitution errors simultaneously plays an important role in DNA-based storage systems as well as in classical communications. This paper deals with the fundamental task of constructing codes that can correct a single insertion or deletion along with a single substitution. A non-asymptotic upper bound on the size of single-deletion single-substitution correcting codes is derived, showing that the redundancy of such a code of length $n$ has to be at least $2 \log n$. The bound is presented both for binary and non-binary codes while an extension to single deletion and multiple substitutions is presented for binary codes. An explicit construction of single-deletion single-substitution correcting codes with at most $6 \log n + 8$ redundancy bits is derived. Note that the best known construction for this problem has to use 3-deletion correcting codes whose best known redundancy is roughly $24 \log n$.