New bounds for the optimal density of covering single-insertion codes via the Tur\'an density

TL;DR

This paper establishes new bounds on the density of covering single-insertion codes, improving previous bounds by connecting code density to Turán density and using extremal combinatorics and measure theory.

Contribution

It introduces tighter bounds on code density by relating it to Turán density, advancing the understanding of optimal code densities in combinatorics.

Findings

Lower bound on code density is improved to 1/r + δ_r

Upper bound on asymptotic code density is improved to approximately 4.911/(r+1)

Bounds are achieved through extremal combinatorics and measure theory techniques

Abstract

We prove that the density of any covering single-insertion code $C\subseteq X^r$ 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 $[0,1]^r$.