Constructing Hypergraphs with Large Girth from Optimal Linear Codes

Belangrijkste concepten

By leveraging constructions from coding theory, we obtain improved lower bounds on the maximum number of edges in r-uniform hypergraphs with girth 5 and 6, for all r ≥ 3.

Samenvatting

The paper studies the maximum number of edges in r-uniform hypergraphs with girth 5 and 6, denoted by exr(N, C<5) and exr(N, C<6) respectively, where N is the number of vertices and C<g denotes the family of Berge cycles of length at most g-1.

Key highlights:

1. The authors address an unproved claim from prior work that the lower bound exr(N, C<5) = Ωr(N^{3/2-o(1)}) holds for all r ≥ 3. They identify an obstacle in the claimed proof and show that this obstacle can be overcome when r ∈ {4, 5, 6}.
2. For all other r, the authors use constructions from coding theory to prove new lower bounds on exr(N, C<5) and exr(N, C<6) that improve upon the previous probabilistic bounds.
3. The authors also show that recent results on hypergraph Turán problems can be used to improve the sphere packing bound for linear codes of distance 6.

The paper provides a comprehensive analysis of the connections between hypergraph Turán problems and coding theory, leading to new insights and improved bounds in both areas.

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Belangrijkste Inzichten Gedestilleerd Uit

Hypergraphs of girth 5 and 6 and coding theory

by Kathryn Haym... om arxiv.org 04-03-2024

https://arxiv.org/pdf/2404.01839.pdf

Diepere vragen

What other connections between hypergraph Turán problems and coding theory could be explored to yield further improvements

One potential avenue for further exploration of the connections between hypergraph Turán problems and coding theory is to investigate the use of algebraic geometric codes. These codes are constructed using algebraic curves and have been shown to have interesting properties related to their minimum distance and error-correcting capabilities. By studying the relationship between hypergraph structures and the properties of algebraic geometric codes, it may be possible to develop new constructions for hypergraphs with improved girth and edge density.

Can the techniques used in this paper be extended to construct hypergraphs with even larger girth

The techniques used in the paper could potentially be extended to construct hypergraphs with even larger girth by adapting the constructions from coding theory to handle more complex structures. For example, exploring the use of concatenated codes or other advanced coding techniques could provide a framework for constructing hypergraphs with girth greater than 6. Additionally, incorporating insights from algebraic geometry and number theory into the construction process may lead to novel approaches for generating hypergraphs with larger girth.

How might the insights from this work on linear codes of distance 6 lead to improvements for linear codes with other distance parameters

The insights gained from the work on linear codes of distance 6 could lead to improvements for linear codes with other distance parameters by generalizing the construction techniques and applying them to codes with different minimum distances. By adapting the parity-check matrix constructions and encoding methods to accommodate different distance requirements, it may be possible to design linear codes with enhanced error-correcting capabilities and improved performance characteristics. Additionally, exploring the relationship between the minimum distance of a code and its structural properties could provide valuable insights for optimizing codes with various distance parameters.