# Anchor-Based Correction of Substitutions in Indexed Sets

## TL;DR

This paper introduces an anchor-based coding scheme for error correction in unordered indexed sets, inspired by DNA data storage, achieving near-optimal redundancy with efficient correction of substitution errors.

## Contribution

It proposes a novel anchoring mechanism for correcting substitutions in unordered sets with indices, enabling the use of standard codes and reducing redundancy.

## Key findings

- Redundancy close to theoretical minimum.
- Less redundancy needed for index correction than data correction.
- Effective correction of substitution errors in unordered sets.

## Abstract

Motivated by DNA-based data storage, we investigate a system where digital information is stored in an unordered set of several vectors over a finite alphabet. Each vector begins with a unique index that represents its position in the whole data set and does not contain data. This paper deals with the design of error-correcting codes for such indexed sets in the presence of substitution errors. We propose a construction that efficiently deals with the challenges that arise when designing codes for unordered sets. Using a novel mechanism, called anchoring, we show that it is possible to combat the ordering loss of sequences with only a small amount of redundancy, which allows to use standard coding techniques, such as tensor-product codes to correct errors within the sequences. We finally derive upper and lower bounds on the achievable redundancy of codes within the considered channel model and verify that our construction yields a redundancy that is close to the best possible achievable one. Our results surprisingly indicate that it requires less redundancy to correct errors in the indices than in the data part of vectors.