Binary $t_1$-Deletion-$t_2$-Insertion-Burst Correcting Codes and Codes Correcting a Burst of Deletions

TL;DR

This paper introduces new binary and non-binary burst error-correcting codes with reduced redundancy, improving existing constructions for correcting deletion and insertion bursts in data transmission.

Contribution

It provides novel constructions of binary and non-binary burst error-correcting codes with lower redundancy than previous methods.

Findings

Redundancy for binary $t_1$-deletion-$t_2$-insertion codes is at most $ ext{log}(n)+(t_1-t_2-1) ext{log} ext{log}(n)+O(1)$.

Improved binary codes correct a burst of 4 deletions with reduced redundancy.

New non-binary $b$-burst-deletion codes with redundancy at most $ ext{log}(n)+(b-1) ext{log} ext{log}(n)+O(1)$.

Abstract

We first give a construction of binary $t_1$-deletion-$t_2$-insertion-burst correcting codes with redundancy at most $\log(n)+(t_1-t_2-1)\log\log(n)+O(1)$, where $t_1\ge 2t_2$. Then we give an improved construction of binary codes capable of correcting a burst of $4$ non-consecutive deletions, whose redundancy is reduced from $7\log(n)+2\log\log(n)+O(1)$ to $4\log(n)+6\log\log(n)+O(1)$. Lastly, by connecting non-binary $b$-burst-deletion correcting codes with binary $2b$-deletion-$b$-insertion-burst correcting codes, we give a new construction of non-binary $b$-burst-deletion correcting codes with redundancy at most $\log(n)+(b-1)\log\log(n)+O(1)$. This construction is different from previous results.