Permutation decoding of codes from generalized Paley graphs

Seneviratne, Padmapani; Limbupasiriporn, Jirapha

doi:10.1007/s00200-013-0198-8

Cited by 8 publications

(4 citation statements)

References 12 publications

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“…Recently, Chi Hoi Yip has studied their clique number ( [32], [33], see also [29]). Both classic Paley graphs and generalized Paley graphs have been used to find linear codes with good decoding properties ( [10], [16], [30]). They can also be seen as particular regular maps in Riemann surfaces ( [13], [14]).…”

Section: Introductionmentioning

confidence: 99%

Generalized Paley graphs equienergetic with their complements

Podestá¹,

Videla²

2022

Preprint

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We consider generalized Paley graphs Γ(k, q), generalized Paley sum graphs Γ + (k, q), and their corresponding complements Γ(k, q) and Γ+ (k, q), for k = 3, 4. Denote by Γ = Γ * (k, q) either Γ(k, q) or Γ + (k, q). We compute the spectra of Γ(3, q) and Γ(4, q) and from them we obtain the spectra of Γ + (3, q) and Γ + (4, q) also. Then we show that, in the non-semiprimitive case, the spectrum of Γ(3, p 3ℓ ) and Γ(4, p 4ℓ ) with p prime can be recursively obtained, under certain arithmetic conditions, from the spectrum of the graphs Γ(3, p) and Γ(4, p) for any ℓ ∈ N, respectively. Using the spectra of these graphs we give necessary and sufficient conditions on the spectrum of Γ * (k, q) such that Γ * (k, q) and Γ * (k, q) are equienergetic for k = 3, 4. In a previous work we have classified all bipartite regular graphs Γ bip and all strongly regular graphs Γsrg which are complementary equienergetic, i.e. {Γ bip , Γbip } and {Γsrg, Γsrg} are equienergetic pairs of graphs. Here we construct infinite pairs of equienergetic non-isospectral regular graphs {Γ, Γ} which are neither bipartite nor strongly regular.

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Section: Introductionmentioning

confidence: 99%

Generalized Paley graphs equienergetic with their complements

Podestá¹,

Videla²

2022

Preprint

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Section: Introductionmentioning

confidence: 99%

“…Generalised Paley graphs have also been investigated in connection with permutation decoding. Earlier work [3,10] on codes derived from the row span of adjacency and incidence matrices of Paley graphs was extended in [15] for all generalised Paley graphs. Finally, Chapter 9.9.1 of the book [8] on graph embeddings in Riemann surfaces deals with generalised Paley maps -the underlying graphs are generalised Paley graphs.…”

Section: Introductionmentioning

confidence: 99%

Generalised Paley Graphs with a Product Structure

Pearce¹,

Praeger

2019

Ann. Comb.

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A graph is Cartesian decomposable if it is isomorphic to a Cartesian product of (more than one) strictly smaller graphs, each of which has more than one vertex and admits no such decomposition. These smaller graphs are called the Cartesian-prime factors of the Cartesian decomposition, and were shown, by Sabidussi and Vizing independently, to be uniquely determined up to isomorphism. We characterise by their parameters those generalised Paley graphs which are Cartesian decomposable, and we prove that for such graphs, the Cartesian-prime factors are themselves smaller generalised Paley graphs. This generalises a result of Lim and the second author which deals with the case where all the Cartesianprime factors are complete graphs. These results contribute to the determination, by parameters, of generalised Paley graphs with automorphism groups larger than the 1-dimensional affine subgroups used to define them.Date: November 2016.

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“…Also, Pearce and Praeger characterized all GP-graphs which are cartesian descomposable ( [12]). Both classic Paley graphs and GP-graphs have been used to find linear codes with good decoding properties ( [2], [7], [17]). They can also be seen as particular regular maps in Riemann surfaces ( [5]).…”

Section: Introductionmentioning

confidence: 99%

A reduction formula for Waring numbers through generalized Paley graphs

Podestá¹,

Videla²

2019

Preprint

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We give a reduction formula for the Waring's number g(k, q) over a finite field F q . By exploiting the relation between g(k, q) with the diameter of the generalized Paley graph Γ(k, q) and by using the characterization of those Γ(k, q) which are cartesian descomposable due to Pearce and Praeger (2016), we obtain the formulafor p prime and a, b, c integers under certain arithmetic conditions. As a particular case, we recover a previous result of Kononen ( 2010) generalizing the result of Winterhof and van de Woestijne (2010). Finally, we use the reduction formula together with the characterization of 2-weight irreducible cyclic codes due to Schmidt and White (2002) to find even values of g(k, q).

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Permutation decoding of codes from generalized Paley graphs

Cited by 8 publications

References 12 publications

Generalized Paley graphs equienergetic with their complements

Generalized Paley graphs equienergetic with their complements

Generalised Paley Graphs with a Product Structure

A reduction formula for Waring numbers through generalized Paley graphs

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