The weight distribution of irreducible cyclic codes associated with decomposable generalized Paley graphs

Podestá, Ricardo A.; Videla, Denis E.

doi:10.3934/amc.2021002

Cited by 6 publications

(4 citation statements)

References 23 publications

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“…The weight distribution of 2-weight irreducible cyclic codes C(k, q) in the semiprimitive case is known (see for instance [26]). Using this and the relation of the weight distribution of C(k, q) with the spectrum of GP-graphs obtained in Theorem 5.4 in [25] one can also recover the spectrum of semiprimitive GP-graphs Γ(k, q) as in (2.7) in Theorem 2.1.…”

Section: 1mentioning

confidence: 82%

The Waring’s problem over finite fields through generalized Paley graphs

Podestá

Videla

2021

Discrete Mathematics

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Section: 1mentioning

confidence: 82%

The Waring’s problem over finite fields through generalized Paley graphs

Podestá

Videla

2021

Discrete Mathematics

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“…We now summarize the main results in the paper. In Section 2, we give the spectrum of Γ * (3, q) and Γ * (4, q) from the spectral relationship between irreducible cyclic codes and GP-graphs (see [21], [23]), and the spectral relationship between Cayley graphs and Cayley sum graphs (see [22]). Namely, in Theorems 2.2 and 2.4 we compute the spectrum of Γ(3, q) and Γ(4, q) using results in [21] while in Theorems 2.8 and 2.9 we obtain, from the spectrum of Γ(3, q) and Γ(4, q), the spectrum of the sum graphs Γ + (3, q) and Γ + (4, q), using results in [22].…”

Section: Cameron's Hierarchymentioning

confidence: 99%

“…Moreover, the diameter of Γ(k, q), if it exists, coincides with the Waring number g(k, q) (see [25], [26]). Under some mild restrictions, the spectrum of GP-graphs determines the weight distribution of their associated irreducible codes ( [21], [23]).…”

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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“…The graphs Γ(3, q) and Γ(4, q) are also of interest (see [19]). The GPgraphs have been extensively studied in the few past years (see for instance [12], [15], [16], [18], [21]. [22], [23], [24].…”

Section: Introductionmentioning

confidence: 99%

Number of cliques of Paley-type graphs over finite commutative local rings

Gallo¹,

Videla²

2023

Preprint

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In this work, given (R, m) a finite commutative local ring with identity and k ∈ N with (k, |R|) = 1, we study the number of cliques of any size in the Cayley graphUsing the known fact that the graph G R (k) can be obtained by blowing-up the vertices of G Fq (k) a number |m| of times, with independence sets the cosets of m, where q is the size of the residue field R/m (see [20]). Then, by using the above blowing-up, we reduce the study of the number of cliques in G R (k) over the local ring R to the computation of the number of cliques of G R/m (k) over the finite residue field R/m ≃ F q . In this way, using known numbers of cliques of generalized Paley graphs (k = 2, 3, 4 and ℓ = 3, 4), we obtain several explicit results for the number of cliques over finite commutative local rings with identity.

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The weight distribution of irreducible cyclic codes associated with decomposable generalized Paley graphs

Cited by 6 publications

References 23 publications

The Waring’s problem over finite fields through generalized Paley graphs

The Waring’s problem over finite fields through generalized Paley graphs

Generalized Paley graphs equienergetic with their complements

Number of cliques of Paley-type graphs over finite commutative local rings

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