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

Oleg Pikhurko, Oleg Verbitsky, Maksim Zhukovskii

TL;DR

The paper develops an analytic framework to bound the density of covering $(r+1)$-insertion codes over large alphabets by linking them to Turán densities. It proves a nontrivial additive lower bound $s(r+1,r)\ge 1/r+\delta_r$ and leverages connections to Turán systems to obtain sharper upper bounds, notably $t(r+1,r)\le 4.911/(r+1)$ for large $r$, which also bound $s(r+1,r)$. The authors establish a fundamental lower bound $s([0,1],r+1,r)\ge 1/r$ using an inverse Bonferroni inequality and subsequently strengthen it via symmetric and structural analysis that yields explicit improvements (e.g., $s(4,3)\ge 0.3333429$). By proving $t(k,r)=s^\ast([0,1],k,r)$ and constructing Turán systems from covering codes, the work deepens the bridge between coding theory and extremal combinatorics and sharpens asymptotic density estimates for large alphabets.

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 $1/r+δ_r$ for some positive real $δ_r$ 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 Turán density from extremal combinatorics. For the last task, we use the analytic framework of measurable subsets of the real cube $[0,1]^r$.

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

TL;DR

The paper develops an analytic framework to bound the density of covering -insertion codes over large alphabets by linking them to Turán densities. It proves a nontrivial additive lower bound and leverages connections to Turán systems to obtain sharper upper bounds, notably for large , which also bound . The authors establish a fundamental lower bound using an inverse Bonferroni inequality and subsequently strengthen it via symmetric and structural analysis that yields explicit improvements (e.g., ). By proving and constructing Turán systems from covering codes, the work deepens the bridge between coding theory and extremal combinatorics and sharpens asymptotic density estimates for large alphabets.

Abstract

We prove that the density of any covering single-insertion code over the -symbol alphabet cannot be smaller than for some positive real not depending on . This improves the volume lower bound of . On the other hand, we observe that, for all sufficiently large , if tends to infinity then the asymptotic upper bound of due to Lenz et al (2021) can be improved to . Both the lower and the upper bounds are achieved by relating the code density to the Turán density from extremal combinatorics. For the last task, we use the analytic framework of measurable subsets of the real cube .
Paper Structure (8 sections, 15 theorems, 55 equations)

This paper contains 8 sections, 15 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Preliminary lemmas
  3. Analytic reformulation
  4. A better lower bound for $s(r+1,r)$
  5. Turán systems
  6. Bounds for small $r$
  7. General bounds
  8. A further improvement of the lower bound for $s(r+1,r)$

Key Result

Theorem 3

For $s(k,r)=\lim_{n\to\infty}S(n,k,r)/n^r$, where the former inequality is true for all $r$ and the latter inequality is true for all sufficiently large $r$.

Theorems & Definitions (25)

  • Definition 1
  • Example 2: Grozea Grozea24
  • Theorem 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Theorem 6
  • proof
  • Theorem 7
  • ...and 15 more