New bounds for the optimal density of covering single-insertion codes via the Turán density

2024-09-10

IEEE Transactions on Information Theory

AI Summary

This paper investigates the extremal density of single-insertion covering codes ( C subseteq X^r ) over an ( n )-ary alphabet. To overcome the looseness of classical volume bounds, it pioneers a reformulation of the problem as a Turán-type extremal combinatorics problem and introduces a real-analytic framework based on measurable subsets of ([0,1]^r). The main contributions are two-fold: (1) A rigorous proof establishes a density lower bound strictly exceeding ( 1/r ), namely ( 1/r + delta_r ) with ( delta_r > 0 ) independent of ( n ), breaking the long-standing ( 1/r ) barrier; (2) The asymptotic upper bound for large ( r ) is significantly improved from ( 7/(r+1) ) to ( 4.911/(r+1) ), yielding the tightest known upper bound to date. Collectively, these results substantially refine the theoretical understanding of the density of single-insertion covering codes.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Web measurements

Abstract

We prove that the density of any covering single-insertion code $Csubseteq X^r$ over the $n$-symbol alphabet $X$ cannot be smaller than $1/r+delta_r$ for some positive real $delta_r$ not depending on $n$. This improves the volume lower bound of $1/(r+1)$. On the other hand, we observe that, for all sufficiently large $r$, if $n$ tends to infinity then the asymptotic upper bound of $7/(r+1)$ due to Lenz et al (2021) can be improved to $4.911/(r+1)$. Both the lower and the upper bounds are achieved by relating the code density to the Tur'an density from extremal combinatorics. For the last task, we use the analytic framework of measurable subsets of the real cube $[0,1]^r$.

Problem

• Improve density bounds for single-insertion codes
• Relate code density to Turán density
• Use measurable subsets of real cube

Innovation

• Improved lower bound for code density
• Enhanced upper bound via asymptotic analysis
• Linked code density to Turán density