t-Deletion-s-Insertion-Burst Correcting Codes

TL;DR

This paper introduces new binary codes capable of correcting complex burst deletion and insertion errors, providing bounds and explicit constructions that optimize redundancy for DNA storage and communication systems.

Contribution

It generalizes burst error correction to include simultaneous deletions and insertions, offering bounds and explicit codes with near-optimal redundancy.

Findings

Sphere-packing upper bound on code size with redundancy at least log n + t - 1.

Explicit construction of codes with redundancy log n + (t-s-1) log log n + O(1).

Constructed a (3,1)-burst correcting code with redundancy at most log n + 9, near optimal.

Abstract

Motivated by applications in DNA-based storage and communication systems, we study deletion and insertion errors simultaneously in a burst. In particular, we study a type of error named $t$-deletion-$s$-insertion-burst ($(t,s)$-burst for short) which is a generalization of the $(2,1)$-burst error proposed by Schoeny {\it et. al}. Such an error deletes $t$ consecutive symbols and inserts an arbitrary sequence of length $s$ at the same coordinate. We provide a sphere-packing upper bound on the size of binary codes that can correct a $(t,s)$-burst error, showing that the redundancy of such codes is at least $\log n+t-1$. For $t\geq 2s$, an explicit construction of binary $(t,s)$-burst correcting codes with redundancy $\log n+(t-s-1)\log\log n+O(1)$ is given. In particular, we construct a binary $(3,1)$-burst correcting code with redundancy at most $\log n+9$, which is optimal up to a constant.