arXiv subjects

Ricardo A. Podestá

Publications and source records attributed to Ricardo A. Podestá.

At least 19 recordsLinked to original sources

arXiv2026

The weight enumerators of irreducible cyclic codes of length $2^\ell$

Using the spectral relation between irreducible cyclic codes (ICC) and generalized Paley (GP) graphs recently found by the authors, we study the weight distribution of irreducible cyclic codes of length $2^\ell$, $\ell \in \mathbb{N}$. These codes were previously studied by Sharma, Bakshi and Raka, who divided them into two families (each having two cases). First we characterize the GP-graphs spectrally related to these families of ICCs: they are complete, semiprimitive, Hamming, or Cartesian products of semiprimitive GP-graphs. We then compute the spectrum of these graphs and hence, by the mentioned spectral relation we finally obtain neat explicit expressions for the weight enumerators of all the irreducible cyclic codes of length $2^\ell$ with $\ell \in \mathbb{N}$.

cs.IT↗

arXiv2026

The spectral relation between irreducible cyclic codes and generalized Paley graphs

Let $p$ be a prime and $\mathbb{F}_q/\mathbb{F}_r$ a finite field extension with $q=p^m$ and $r=p^s$. For any $k\mid q-1$, we consider $r$-ary irreducible cyclic codes (ICC) of the form $C(k,q/r) = \{(Tr_{q/r}(γω^{ik})_{i=0}^{n-1})\}_{γ\in \mathbb{F}_r}$, with $ω$ a primitive element of $\mathbb{F}_q$ and $ n= \tfrac{q-1}{k}$, and generalized Paley (GP) graphs $Γ(k,q) = Cay(\mathbb{F}_q, \{ x^k : x \in \mathbb{F}_q^* \})$. We show that there is a simple closed formula relating the weight distribution of $C(k,q/r)$ with the spectrum of $Γ(k_r,q)$, where $k_r=\gcd(k, \frac{q-1}{r-1})$. Then, we give $Spec(Γ(k,q))$ explicitly for those graphs associated with irreducible 2-weight cyclic codes in the semiprimitive and exceptional cases. Finally, we give the weight enumerators of irreducible cyclic codes associated with Hamming GP-graphs.

cs.IT↗

arXiv2026

The energy of a family of mirror di-Cayley (sum) graphs: equienergy and moments

For a group $G$ and subsets $S,T \subset G$ we consider a family of mirror di-Cayley graphs $MX(G;S,T)$ and mirror di-Cayley sums graphs $MX^+(G;S,T)$, namely those with $T=\{e\}, S$ or $S \cup \{e\}$. We refer to them indistinctly by $MX^*(G;S,T)$. We can think of $MX^*(G;S,T)$ as two copies of the Cayley (sum) graph $X^*(G,S)$ joined by edges determined by the connection set $T$. Recently, in the work \textit{Isospectral Cayley graphs with even and odd spectrum}, we study the spectrum of these graphs and several isospectrality problems. Here, we compute the energy and spectral $k$-moments of $MX^*(G;S,T)$ in terms of those of the underlying Cayley graphs $X(G,S)$. Then, we study energetic problems like hypo-, order-, and hyper-energeticity of these graphs. Finally, we give conditions for the existence of equienergetic pairs of non-isomorphic MDCGs.

math.CO↗

arXiv2026

On the number of solutions of systems of diagonal equations through diagonal GP-graphs: the general and the Hermitian-form cases

For any $m, s \in \mathbb{N}$, we study the number $N_{m\times s,q}(κ, β)$ of solutions $(x_1,\ldots,x_s) \in (\mathbb{F}_q)^s$ of the monic system of diagonal equations $$ X_{1}^{k_i} + \cdots + X_{s}^{k_i}= β_i, \qquad (1\le i \le m), $$ with $κ=(k_1,\ldots,k_m) \in \mathbb{N}^m$ and $β=(β_1,\ldots,β_m) \in (\mathbb{F}_q)^m$. We show that this number can be obtained in terms of some data of \textit{diagonal} GP-graphs $Γ(κ,q)$. This is a new family of graphs that we introduce here, $Γ(κ,q)$, with $κ= (k_1,\ldots,k_m) \in \mathbb{N}^{m}$, is the directed graph with vertex set the finite field $\mathbb{F}_q$ and there is an arc from $u$ to $v$ if and only if $v-u \in R_κ = \{ (x^{k_1},\ldots,x^{k_m}) : x \in \mathbb{F}_{q}^*\}$. In particular, we give three different expressions for $N_{m\times s,q}(κ, β)$: one in terms of walks, another in terms of adjacency matrices of $Γ(κ,q)$ and the last one in terms of the spectrum of $Γ(κ,q)$. Finally, we explicitly derive combinatorial formulas for the number of solutions $N_{m}(s,q) = N_{m\times s,q}(κ_\ell, 0)$ of monic homogeneous systems of diagonal equations of the form $$ X_1^{q^{\ell_i}+1} + \cdots + X_s^{q^{\ell_i}+1} = 0 \qquad (1\le i \le m),$$ with $κ_\ell=(\ell_1,\ldots,\ell_m)=(1,3,\ldots,2m-1)$ and $m\ge 2$, via the known spectrum of Hermitian-form graphs, which can be viewed as diagonal GP-graphs. For any $m,s \in \mathbb{N}$, we give general summation and recursive formulas for $N_m(s,q) \in \mathbb{Z}[q]$. For the small cases $N_{1}(s,q)$, $N_{2}(s,q)$ and $N_{m}(s,q)$, with $1\le s \le 5$, we give explicit expressions.

math.CO↗

arXiv2026

On $k$-th unitary Cayley graphs over finite commutative rings: structure and decompositions

Given $R$ a finite commutative ring with identity and $k \in \mathbb{N}$, we consider the $k$-th unitary Cayley graph $G_R(k)=Cay(R,U_{R,k})$ with $U_{R,k} = \{ x^k: x \in R^*\}$, and its symmetrized version $\mathcal{G}_R(k) = Cay(R,T_{R,k})$, with $T_{R,k}=U_{R,k} \cup (-U_{R,k})$. If $R$ is a local ring with maximal ideal $\frak m$, we give the blow-up decompositions for the graphs: namely, we have $G_R(k)= (G_{R/\frak m}(k))^{(|\frak m|)}$ and $\mathcal{G}_R(k)= (\mathcal{G}_{R/\frak m}(k))^{(|\frak m|)}$ for any $k$ such that $(k,|R|)=1$. If the ring $R$ has Artin decomposition $R=R_1 \times \cdots \times R_s$ in local rings $R_i$, we give the Kronecker product decompositions $G_R(k) = G_{R_1}(k) \otimes \cdots \otimes G_{R_s}(k)$ and $\mathcal{G}_R(k) = \mathcal{G}_{R_1}(k) \otimes \cdots \otimes \mathcal{G}_{R_s}(k)$. In further $(k,|R|)=1$, these decompositions can be given in terms of generalized Paley (GP) graphs over finite fields, that is $G_{R_i}(k) = Γ(k_i,q_i)$ and similarly for $\mathcal{G}_{R_i}(k)$, for $i=1,\ldots,s$. Also, the reduced graphs correspond to the graphs of the reduced rings, i.e.\@ $\big(G_{R}(k)\big)_{red}\simeq G_{R_{red}}(k)$ and $\big(\mathcal{G}_{R}(k)\big)_{red} \simeq \mathcal{G}_{R_{red}}(k)$. By using these decompositions in terms of GP-graphs, we study some basic structural properties of the graphs such as directedness, bipartiteness and connectedness.

math.CO↗

arXiv2026

Isospectral Cayley graphs with even and odd spectrum

For a group $G$ and subsets $S,T \subset G$ we introduce the mirror di-Cayley graph $MX(G;S,T)$ and mirror di-Cayley sum graph $MX^+(G;S,T)$ with connections sets $S$ and $T$ (MDCGs for short). We refer to them indistinctly by $MX^*(G;S,T)$. We then consider the family $\mathcal{F}$ of those MDCGs with $T \in \mathcal{S}$, where $\mathcal{S}= \big\{ \{e\}, S, S \cup \{e\} \big\}$. We compute the spectra of the graphs $MX^*(G;S,T)$, with $T \in \mathcal{S}$, in terms of those of the corresponding Cayley graphs $X^*(G,S)$. We show that if $X(G,S)$ has integral spectrum then $MX^*(G;S,T)$ is also integral for any $T \in \mathcal{S}$, but $MX^*(G;S,S)$ has even spectrum (all even eigenvalues) and $MX^*(G;S,S \cup \{e\})$ has odd spectrum (all odd eigenvalues), an interesting phenomenom which seems to be new. We then study isospectrality between different pairs of MDCGs in terms of the isospectrality of the underlying Cayley graphs. Finally, using unitary Cayley graphs $X(R,R^*)$ over a finite commutative ring $R$, which is known to be integral, we construct pairs of integral isospectral mirror di-Cayley (sum) graphs $\{ MX(R;R^*, T), MX^+(R;R^*, T) \}$, both with even (resp.\@ odd) spectrum for $T=R^*$ (resp.\@ $T=R^* \cup \{0\}$). All these examples can be seen as Cayley (sum) graphs over $G=R \times \mathbb{Z}_2$, hence obtaining pairs of even and odd isospectral Cayley graphs of the form $\{Γ, Γ^+\}$.

math.CO↗

arXiv2026

The nature of the spectrum of generalized Paley graphs and weak Waring numbers over finite fields

We consider the family of generalized Paley graphs (GP-graphs for short) $Γ(k,q) = Cay(\mathbb{F}_q, (\mathbb{F}_q^*)^k)$, with $q=p^m$ and $p$ prime. We characterize all GP-graphs having real spectrum; namely, $Spec(Γ(k,q)) \subset \mathbb{R}$ if and only if $Γ(k,q)$ is undirected. We then study conditions for integrality in the spectrum and give a general method to produce integral GP-graphs through cyclotomic polynomials. Using this, we construct several infinite families of integral GP-graphs. Next, we focus on directed GP-graphs (GP-digraphs). We show that GP-digraphs always have three or more eigenvalues, and then we prove that there is only one kind of GP-digraphs having three different eigenvalues: the oriented Paley graphs $\vec{\mathcal{P}}_q$ or disjoint unions of copies of them, $\vec{\mathcal{P}}_q \cup \cdots \cup \vec{\mathcal{P}}_q$. Then, we show that generically the GP-digraphs have period 1 (equivalently index of imprimitivity 1) except for $Γ(q-1,q)$ with $q$ odd, which is the disjoint union of oriented $p$-cycles, having period $p$. Finally, as an application, we study weak Waring numbers over finite fields through GP-graphs. In particular, we reduce the computation of the weak Waring numbers over finite fields to the computation of classic Waring numbers over finite fields, a result previously obtained by Cochrane and Cipra in 2012 by other means.

math.CO↗

arXiv2025

Connected components and non-bipartiteness of generalized Paley graphs

In this work we consider the class of Cayley graphs known as generalized Paley graphs (GP-graphs for short) given by $Γ(k,q) = Cay(\mathbb{F}_q, \{x^k : x\in \mathbb{F}_q^* \})$, where $\mathbb{F}_q$ is a finite field with $q$ elements, both in the directed and undirected case. Hence $q=p^m$ with $p$ prime, $m\in \mathbb{N}$ and one can assume that $k\mid q-1$. We first give the connected components of an arbitrary GP-graph. We show that these components are smaller GP-graphs all isomorphic to each other (generalizing a Lim and Praeger's result from 2009 to the directed case). We then characterize those GP-graphs which are disjoint unions of odd cycles. Finally, we show that $Γ(k,q)$ is non-bipartite except for the graphs $Γ(2^m-1,2^m)$, $m \in \mathbb{N}$, which are isomorphic to $K_2 \sqcup \cdots \sqcup K_2$, the disjoint union of $2^{m-1}$ copies of $K_2$.

math.CO↗

arXiv2025

Spectral properties of generalized Paley graphs

We study the spectrum of generalized Paley graphs $Γ(k,q)=Cay(\mathbb{F}_q,R_k)$, undirected or not, with $R_k=\{x^k:x\in \mathbb{F}_q^*\}$ where $q=p^m$ with $p$ prime and $k\mid q-1$. We first show that the eigenvalues of $Γ(k,q)$ are given by the Gaussian periods $η_{i}^{(k,q)}$ with $0\le i\le k-1$. Then, we explicitly compute the spectrum of $Γ(k,q)$ with $1\le k \le 4$ and of $Γ(5,q)$ for $p\equiv 1\pmod 5$ and $5\mid m$. Also, we characterize those GP-graphs having integral spectrum, showing that $Γ(k,q)$ is integral if and only if $p$ divides $(q-1)/(p-1)$. Next, we focus on the family of semiprimitive GP-graphs. We show that they are integral strongly regular graphs (of pseudo-Latin square type). Finally, we characterize all integral Ramanujan graphs $Γ(k,q)$ with $1\le k \le 4$ or where $(k,q)$ is a semiprimitive pair.

math.CO↗

arXiv2024

The spectra of generalized Paley graphs of $(q^\ell+1)$-th powers and applications

We consider a special class of generalized Paley graphs over finite fields, namely the Cayley graphs with vertex set $\mathbb{F}_{q^m}$ and connection set the nonzero $(q^\ell+1)$-th powers in $\mathbb{F}_{q^m}$, as well as their complements. We explicitly compute the spectrum and the energy of these graphs. As a consequence, the graphs turn out to be (with trivial exceptions) simple, connected, non-bipartite, integral and strongly regular, of pseudo or negative Latin square type. By using the spectral information we compute several invariants of these graphs. We exhibit infinitely many pairs of equienergetic non-isospectral graphs. As applications, on the one hand we solve Waring's problem over $\mathbb{F}_q^m$ for the exponents $q^\ell+1$, for each $q$ and for infinitely many values of $\ell$ and $m$. We obtain that the Waring's number $g(q^\ell+1,q^m)=1$ or $2$, depending on $m$ and $\ell$, thus solving some open cases. On the other hand, we construct infinite towers of Ramanujan graphs in all characteristics. Finally, we give the Ihara zeta functions of these graphs.

math.CO↗

arXiv2023

Waring numbers over finite commutative local rings

In this paper we study Waring numbers $g_R(k)$ for $(R,\frak m)$ a finite commutative local ring with identity and $k \in \mathbb{N}$ with $(k,|R|)=1$. We first relate the Waring number $g_R(k)$ with the diameter of the Cayley graphs $G_R(k)=Cay(R,U_R(k))$ and $W_R(k)=Cay(R,S_R(k))$ with $U_R(k) = \{ x^k : x\in R^*\}$ and $S_R(k)=\{x^k : x\in R^\times\}$, distinguishing the cases where the graphs are directed or undirected. We show that in both cases (directed or undirected), the graph $G_R(k)$ can be obtained by blowing-up the vertices of $G_{\mathbb{F}_q}(k)$ a number $|\frak{m}|$ of times, with independence sets the cosets of $\frak{m}$, where $q$ is the size of the residue field $R/\frak m$. Then, by using the above blowing-up, we reduce the study of the Waring number $g_R(k)$ over the local ring $R$ to the computation of the Waring number $g(k,q)$ over the finite residue field $R/\frak m \simeq \mathbb{F}_q$. In this way, using known results for Waring numbers over finite fields, we obtain several explicit results for Waring numbers over finite commutative local rings with identity.

math.AC↗

arXiv2023

Lee metrics on groups

In this work we consider interval metrics on groups; that is, integral invariant metrics whose associated weight functions do not have gaps. We give conditions for a group to have and to have not interval metrics. Then we study Lee metrics on general groups, that is interval metrics having the finest unitary symmetric associated partition. These metrics generalize the classic Lee metric on cyclic groups. In the case that $G$ is a torsion-free group or a finite group of odd order, we prove that $G$ has a Lee metric if and only if $G$ is cyclic. Also, if $G$ is a group admitting Lee metrics then $G \times \mathbb{Z}_2^k$ always have Lee metrics for every $k \in \mathbb{N}$. Then, we show that some families of metacyclic groups, such as cyclic, dihedral, and dicyclic groups, always have Lee metrics. Finally, we give conditions for non-cyclic groups such that they do not have Lee metrics. We end with tables of all groups of order $\le 31$ indicating which of them have (or have not) Lee metrics and why (not).

math.GR↗

arXiv2022

A reduction formula for Waring numbers through generalized Paley graphs

We give a reduction formula for the Waring number $g(k,q)$ over a finite field $\mathbb{F}_q$. By exploiting the relation between $g(k,q)$ with the diameter of the generalized Paley graph $Γ(k,q)$ and by using the characterization due to Pearce and Praeger (2019) of those $Γ(k,q)$ which are Cartesian decomposable, we obtain the reduction formula $$g(\tfrac{p^{ab}-1}{bc},p^{ab}) = b g(\tfrac{p^a-1}{c},p^a)$$ for $p$ prime and $a,b,c$ positive integers under certain arithmetic conditions. Then, we find some arithmetic conditions to apply the formula above, which allow us to obtain many infinite families of explicit values of Waring numbers. Finally, we use the reduction formula together with the characterization of $2$-weight irreducible cyclic codes due to Schmidt and White (2002) to find infinite families of explicit even values of $g(k,q)$.

math.NT↗

arXiv2022

Generalized Paley graphs equienergetic with their complements

We consider generalized Paley graphs $Γ(k,q)$, generalized Paley sum graphs $Γ^+(k,q)$, and their corresponding complements $\bar Γ(k,q)$ and $\bar Γ^+(k,q)$, 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 $Γ(3,p^{3\ell})$ and $Γ(4,p^{4\ell})$ 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 $\ell \in \mathbb{N}$, 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 $\bar Γ^*(k,q)$ are equienergetic for $k=3,4$. In a previous work we have classified all bipartite regular graphs $Γ_{bip}$ and all strongly regular graphs $Γ_{srg}$ which are complementary equienergetic, i.e.\@ $\{Γ_{bip}, \barΓ_{bip}\}$ and $\{Γ_{srg}, \barΓ_{srg}\}$ are equienergetic pairs of graphs. Here we construct infinite pairs of equienergetic non-isospectral regular graphs $\{Γ, \bar Γ\}$ which are neither bipartite nor strongly regular.

math.CO↗

arXiv2021

Invariant metrics on finite groups

We study invariant and bi-invariant metrics on groups focusing on finite groups $G$. We show that non-equivalent (bi) invariant metrics on $G$ are in 1-1 correspondence with unitary symmetric (conjugate) partitions on $G$. To every metric group $(G,d)$ we associate to it the symmetry group and the weighted graph of distances. Using these objects we can classify all equivalence classes of invariant and bi-invariant metrics for small groups. We then study the number of non-equivalent invariant and bi-invariant metrics on $G$. We give an expression for the number of such metrics in terms of Bell numbers, with closed expressions for certain groups such as abelian, dihedral, quasidihedral and dicyclic groups. We then characterize all the groups (finite or not) in which every invariant metric is also bi-invariant. We give the number of non-equivalent invariant and bi-invariant metrics for all the groups of order up to 32.

math.CO↗

arXiv2021

The conorm code of an AG-code

Given a suitable extension $F'/F$ of algebraic function fields over a finite field $\mathbb{F}_q$, we introduce the conorm code $\text{Con}_{F'/F}(\mathcal{C})$ defined over $F'$ which is constructed from an algebraic geometry code $\mathcal{C}$ defined over $F$. We study the parameters of $\text{Con}_{F'/F}(\mathcal{C})$ in terms of the parameters of $\mathcal{C}$, the ramification behavior of the places used to define $\mathcal{C}$ and the genus of $F$. In the case of unramified extensions of function fields we prove that $\text{Con}_{F'/F}(\mathcal{C})^\perp = \text{Con}_{F'/F}({\mathcal{C}}^\perp)$ when the degree of the extension is coprime to the characteristic of $\mathbb{F}_q$. We also study the conorm of cyclic algebraic-geometry codes and we show that some repetition codes, Hermitian codes and all Reed-Solomon codes can be represented as conorm codes.

math.NT↗

arXiv2021

On cyclic algebraic-geometry codes

In this paper we initiate the study of cyclic algebraic geometry codes. We give conditions to construct cyclic algebraic geometry codes in the context of algebraic function fields over a finite field by using their group of automorphisms. We prove that cyclic algebraic geometry codes constructed in this way are closely related to cyclic extensions. We also give a detailed study of the monomial equivalence of cyclic algebraic geometry codes constructed with our method in the case of a rational function field.

math.AG↗

arXiv2021

On regular graphs equienergetic with their complements

We give necessary and sufficient conditions on the parameters of a regular graph $Γ$ (with or without loops) such that $E(Γ)=E(\overline Γ)$. We study complementary equienergetic cubic graphs obtaining classifications up to isomorphisms for connected cubic graphs with single loops (5 non-isospectral pairs) and connected integral cubic graphs without loops ($Γ= K_3 \square K_2$ or $Q_3$). Then we show that, up to complements, the only bipartite regular graphs equienergetic and non-isospectral with their complements are the crown graphs $Cr(n)$ or $C_4$. Next, for the family of strongly regular graphs $Γ$ we characterize all possible parameters $srg(n,k,e,d)$ such that $E(Γ) = E(\overline Γ)$. Furthermore, using this, we prove that a strongly regular graph is equienergetic to its complement if and only if it is either a conference graph or else it is a pseudo Latin square graph (i.e. has $OA$ parameters). We also characterize all complementary equienergetic pairs of graphs of type $\mathcal{C}(2)$, $\mathcal{C}(3)$ and $\mathcal{C}(5)$ in Cameron's hierarchy (the cases $\mathcal{C}(1)$ and $\mathcal{C}(4)$ are still open). Finally, we consider unitary Cayley graphs over rings $G_R=X(R,R^*)$. We show that if $R$ is a finite Artinian ring with an even number of local factors, then $G_R$ is complementary equienergetic if and only if $R=\mathbb{F}_q \times \mathbb{F}_{q'}$ is the product of 2 finite fields.

math.CO↗