Minimum supports of eigenfunctions of graphs: a survey

Sotnikova, E. V.; Valyuzhenich, Alexandr

doi:10.26493/2590-9770.1404.61e

Cited by 5 publications

(4 citation statements)

References 74 publications

(135 reference statements)

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“…More generally, one can ask what is the minimum difference between two MRD codes of the same parameters. This question is common for many classes of optimal codes, and one of the developed methods to make lower bounds for this minimum is to study eigenspaces of the ambient graph, see [24] for a survey for different graphs and [25] for the bilinear form graphs.…”

Section: Discussionmentioning

confidence: 99%

Constructing MRD codes by switching

Shi¹,

Krotov²,

Özbudak³

2022

Preprint

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MRD codes are maximum codes in the rank-distance metric space on m-by-n matrices over the finite field of order q. They are diameter perfect and have the cardinality q m(n−d+1) if m ≥ n. We define switching in MRD codes as replacing special MRD subcodes by other subcodes with the same parameters. We consider constructions of MRD codes admitting such switching, including punctured twisted Gabidulin codes and direct-product codes. Using switching, we construct a huge class of MRD codes whose cardinality grows doubly exponentially in m if the other parameters (n, q, the code distance) are fixed. Moreover, we construct MRD codes with different affine ranks and aperiodic MRD codes.

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Section: Discussionmentioning

confidence: 99%

Constructing MRD codes by switching

Shi¹,

Krotov²,

Özbudak³

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…More generally, one can ask what is the minimum difference between two MRD codes of the same parameters. This question is common for many classes of optimal codes, and one of the developed methods to make lower bounds for this minimum is to study eigenspaces of the ambient graph, see [32] for a survey for different graphs and [33] for the bilinear form graphs.…”

Section: Codes With Different Affine Ranksmentioning

confidence: 99%

Constructing MRD codes by switching

Shi,

Krotov,

Özbudak

2024

J of Combinatorial Designs

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Maximum rank‐distance (MRD) codes are (not necessarily linear) maximum codes in the rank‐distance metric space on ‐by‐ matrices over a finite field . They are diameter perfect and have the cardinality if . We define switching in MRD codes as the replacement of special MRD subcodes by other subcodes with the same parameters. We consider constructions of MRD codes admitting switching, such as punctured twisted Gabidulin codes and direct‐product codes. Using switching, we construct a huge class of MRD codes whose cardinality grows doubly exponentially in if the other parameters (, the code distance) are fixed. Moreover, we construct MRD codes with different affine ranks and aperiodic MRD codes.

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“…Although eigenfunctions of graphs receive less attention of researchers in contrast to their eigenvalues, there are still tons of related literature. We refer to the recent survey [22] for a summary of results on the problem of finding the minimum cardinality of support of eigenfunctions of graphs and characterising the optimal eigenfunctions. The following lemma gives a lower bound for the number of non-zeroes (i.e., the cardinality of the support) for an eigenfunction of a strongly regular graph.…”

Section: The Weight-distribution Bound For Strongly Regular Graphsmentioning

confidence: 99%

On eigenfunctions and maximal cliques of generalised Paley graphs of square order

Goryaĭnov¹,

Shalaginov²,

Yip³

2022

Preprint

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Let GP (q 2 , m) be the m-Paley graph defined on the finite field with order q 2 . We study eigenfunctions and maximal cliques in generalised Paley graphs GP (q 2 , m), where m | (q+1). In particular, we explicitly construct maximal cliques of size q+1 m or q+1 m + 1 in GP (q 2 , m), and show the weight-distribution bound on the cardinality of the support of an eigenfunction is tight for the smallest eigenvalue − q+1 m of GP (q 2 , m). These new results extend the work of Baker et. al and Goryainov et al. on Paley graphs of square order. We also study the stability of the Erdős-Ko-Rado theorem for GP (q 2 , m) (first proved by Sziklai).

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Minimum supports of eigenfunctions of graphs: a survey

Abstract: In this work we present a survey of results on the problem of finding the minimum cardinality of the support of eigenfunctions of graphs.

Cited by 5 publications

References 74 publications

Constructing MRD codes by switching

Constructing MRD codes by switching

Constructing MRD codes by switching

On eigenfunctions and maximal cliques of generalised Paley graphs of square order

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