New Bounds for the Optimal Density of Covering SingleInsertion Codes Using Turán Density
Core Concepts
The density of any covering singleinsertion code over an nsymbol alphabet cannot be smaller than 1/r + δr for some positive real δr not depending on n, improving the previous lower bound of 1/(r+1). The asymptotic upper bound can also be improved from 7/(r+1) to 4.911/(r+1) for sufficiently large r.
Abstract
The paper establishes new bounds for the optimal density of covering singleinsertion codes by relating the code density to the Turán density from extremal combinatorics.
The key highlights are:

A lower bound is proved showing that the density of any covering singleinsertion code C over an nsymbol alphabet X cannot be smaller than 1/r + δr for some positive real δr not depending on n. This improves the previous lower bound of 1/(r+1).

An upper bound is derived showing that for all sufficiently large r, if n tends to infinity then the asymptotic upper bound of the code density can be improved from 7/(r+1) to 4.911/(r+1).

Both the lower and upper bounds are achieved by relating the code density to the Turán density, which is a wellstudied concept in extremal combinatorics. An analytic framework of measurable subsets of the real cube [0,1]^r is used for this purpose.

A reverse relation between the optimal density of covering singleinsertion codes and the Turán density is established, allowing the lower bound to be further improved.
The paper provides a comprehensive analysis of the optimal density of covering singleinsertion codes, significantly improving the previously known bounds.
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New bounds for the optimal density of covering singleinsertion codes via the Tur\'an density
Stats
The density of any covering singleinsertion code C over an nsymbol alphabet X cannot be smaller than 1/r + δr for some positive real δr not depending on n.
For all sufficiently large r, if n tends to infinity then the asymptotic upper bound of the code density can be improved from 7/(r+1) to 4.911/(r+1).
Quotes
"We prove that the density of any covering singleinsertion code C ⊆ X^r over the nsymbol alphabet X cannot be smaller than 1/r + δr for some positive real δr not depending on n."
"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)."
Deeper Inquiries
How can the techniques developed in this paper be extended to analyze the optimal density of covering codes for other types of edit distance metrics, such as deletions or transpositions?
The techniques developed in this paper, particularly the use of Turán density and the analytic framework involving measurable subsets, can be adapted to analyze covering codes for other edit distance metrics, such as deletions and transpositions. For instance, the concept of covering codes can be generalized to account for sequences that allow for deletions by defining a new type of covering code that considers the subsequences formed by removing elements from a sequence.
In the case of transpositions, one could define a covering code that allows for the rearrangement of elements within a sequence. This would involve extending the definitions of covering codes to include permutations of sequences, thereby creating a new class of codes that can cover sequences under the transposition metric.
The analytic methods employed in this paper, such as the use of measurable sets and the application of Bonferroni inequalities, can be similarly applied to derive bounds for these new types of codes. By establishing relationships between the densities of these codes and their corresponding Turán densities, researchers can explore the optimal densities for covering codes under various edit distance metrics, leading to a deeper understanding of their combinatorial properties.
What are the implications of the improved bounds on the practical applications of covering singleinsertion codes, such as in data compression or errorcorrecting codes?
The improved bounds on the optimal density of covering singleinsertion codes have significant implications for practical applications in data compression and errorcorrecting codes. In data compression, the ability to efficiently cover sequences with minimal redundancy is crucial. The new lower bound of ( \frac{1}{r} + \delta_r ) indicates that it is possible to achieve a more compact representation of data while still maintaining the ability to recover original sequences after single insertions. This can lead to more efficient algorithms for lossless compression, where the goal is to minimize the size of the data without losing any information.
In the realm of errorcorrecting codes, the improved upper bound of ( \frac{4.911}{r + 1} ) suggests that the design of codes can be optimized to correct errors more effectively while using fewer resources. This is particularly important in communication systems where bandwidth is limited, and the ability to transmit data reliably is paramount. The findings in this paper can guide the development of new coding schemes that are both efficient and robust against errors, enhancing the reliability of data transmission in various applications, including telecommunications and storage systems.
Are there any connections between the optimal density of covering singleinsertion codes and other fundamental problems in combinatorics or computer science?
Yes, there are several connections between the optimal density of covering singleinsertion codes and other fundamental problems in combinatorics and computer science. One notable connection is with the study of Turán systems, which are closely related to extremal graph theory. The bounds established for covering singleinsertion codes leverage the properties of Turán density, indicating a deep interplay between these areas.
Additionally, the concepts of covering codes are linked to the theory of errorcorrecting codes, which is a central topic in coding theory. The optimization of covering codes can inform the design of codes that correct various types of errors, including insertions, deletions, and substitutions, which are common in practical applications.
Moreover, the techniques used to analyze covering codes, such as measurable sets and probabilistic methods, are also applicable in other areas of combinatorial optimization and algorithm design. For example, the use of probabilistic methods to derive bounds can be seen in random graph theory and the analysis of randomized algorithms, highlighting the broader relevance of the findings in this paper to various combinatorial and computational problems.
Overall, the study of covering singleinsertion codes not only advances our understanding of coding theory but also enriches the broader field of combinatorial optimization and its applications in computer science.