Codes Correcting Two Bursts of Exactly $b$ Deletions

TL;DR

This paper introduces new $q$-ary codes capable of correcting two bursts of exactly $b$ deletions with reduced redundancy, improving upon previous constructions for all $b>1$ and $q eq 2$.

Contribution

The work presents a novel construction method for $q$-ary codes that correct two deletion bursts with lower redundancy than prior syndrome compression techniques.

Findings

Redundancy reduced to $5 ext{log} n + O( ext{log} ext{log} n)$ bits for all $b>1$ and $q eq 2$.

New codes for $b=1$ case with similar redundancy bounds for all $q>2$.

Improved code constructions outperform previous best known methods.

Abstract

In this paper, we investigate codes designed to correct two bursts of deletions, where each burst has a length of exactly $b$, where $b>1$. The previous best construction, achieved through the syndrome compression technique, had a redundancy of at most $7\log n+O\left(\log n/\log\log n\right)$ bits. In contrast, our work introduces a novel approach for constructing $q$-ary codes that attain a redundancy of at most $5\log n+O(\log\log n)$ bits for all $b>1$ and $q\ge2$. Additionally, for the case where $b=1$, we present a new construction of $q$-ary two-deletion correcting codes with a redundancy of $5\log n+O(\log\log n)$ bits, for all $q>2$.