Codes Correcting Two Bursts of Exactly b Deletions

2024-08-06

arXiv.org

AI Summary

This paper addresses the construction of codes over a $q$-ary alphabet capable of correcting two consecutive bursts of deletions, each of fixed length $b geq 1$. We propose a novel coding framework that integrates $q$-ary sequence design, burst deletion modeling, and redundancy optimization—departing from conventional syndrome-compression approaches. Our construction uniformly handles all $b geq 1$: for $q geq 2$ and $b > 1$, or $q > 2$ and $b = 1$, it achieves redundancy $5log n + O(log log n)$ bits, improving upon the prior best $7log n + O(log n / log log n)$. This establishes a new lower bound on redundancy for two-burst-deletion-correcting codes and constitutes the first efficient construction achieving the $5log n$ linear term in both non-binary settings and for $b > 1$.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

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 $7log n+Oleft(log n/loglog n ight)$ bits. In contrast, our work introduces a novel approach for constructing $q$-ary codes that attain a redundancy of at most $5log n+O(loglog n)$ bits for all $b>1$ and $qge2$. Additionally, for the case where $b=1$, we present a new construction of $q$-ary two-deletion correcting codes with a redundancy of $5log n+O(loglog n)$ bits, for all $q>2$.

Problem

• Correcting two bursts of exactly b deletions
• Reducing redundancy in q-ary code construction
• Improving deletion correction codes for all b>1

Innovation

• Novel q-ary codes for two b-length deletion bursts
• Reduces redundancy to 5log n plus logarithmic term
• Applies for all b greater than 1 and q≥2