Codes in the Space of Multisets---Coding for Permutation Channels with Impairments

TL;DR

This paper investigates error-correcting codes for multisets in permutation channels with impairments like insertions, deletions, and substitutions, providing optimal constructions and bounds for such codes.

Contribution

It introduces new code constructions for multisets under complex errors and proves the optimality of a Sidon set-based construction across various parameters.

Findings

Sidon set-based codes are asymptotically optimal for error correction.

Bounds on the size of optimal codes are established.

The proposed codes are maximal in certain cases.

Abstract

Motivated by communication channels in which the transmitted sequences are subject to random permutations, as well as by certain DNA storage systems, we study the error control problem in settings where the information is stored/transmitted in the form of multisets of symbols from a given finite alphabet. A general channel model is assumed in which the transmitted multisets are potentially impaired by insertions, deletions, substitutions, and erasures of symbols. Several constructions of error-correcting codes for this channel are described, and bounds on the size of optimal codes correcting any given number of errors derived. The construction based on the notion of Sidon sets in finite Abelian groups is shown to be optimal, in the sense of the asymptotic scaling of code redundancy, for any "error radius" and any alphabet size. It is also shown to be optimal in the stronger sense of maximal code cardinality in various cases.