arXiv subjects
Publications and source records attributed to Ricardo A. Podestá.
We use known characterizations of generalized Paley graphs which are cartesian decomposable to explicitly compute the spectra of the corresponding associated irreducible cyclic codes. As applications, we give reduction formulas for the number of rational points in Artin-Schreier curves defined over extension fields and to the computation of Gaussian periods.
We show that the Waring's number over a finite field $\mathbb{F}_q$, denoted $g(k,q)$, when exists, coincides with the diameter of the generalized Paley graph $Γ(k,q)=Cay(\mathbb{F}_{q},R_k)$ with $R_k=\{x^k : x\in \mathbb{F}_q^*\}$. We find infinite new families of exact values of $g(k,q)$ from a characterization of graphs $Γ(k,q)$ which are also Hamming graphs previously proved by Lim and Praeger in 2009. Then, we show that every positive integer is the Waring number for some pair $(k,q)$ with $q$ not a prime. Finally, we find a lower bound for $g(k,p)$ with $p$ prime by using that $Γ(k,p)$ is a circulant graph in this case.
We prove that the Cayley graphs $X(G,S)$ and $X^+(G,S)$ are equienergetic for any abelian group $G$ and any symmetric subset $S$. We then focus on the family of unitary Cayley graphs $G_R=X(R,R^*)$, where $R$ is a finite commutative ring with identity. We show that under mild conditions, $\{G_R, G_R^+\}$ are pairs of integral equienergetic non-isospectral graphs (generically connected and non-bipartite). Then, we obtain conditions such that $\{G_R, \bar G_R\}$ are equienergetic non-isospectral graphs. Finally, we characterize all integral equienergetic non-isospectral triples $\{G_R, G_R^+, \bar G_R \}$ such that all the graphs are also Ramanujan.
We prove that if $H$ is a subgroup of index $n$ of any cyclic group $G$, then $G$ can be isometrically embedded in $(H^n, d_{_{Ham}}^n)$, thus generalizing previous results of Carlet (1998) for $G=\mathbb{Z}_{2^k}$ and Yildiz-Özger (2012) for $G=\mathbb{Z}_{p^k}$ with $p$ prime. Next, for any positive integer $q$ we define the $q$-adic metric $d_q$ in $\mathbb{Z}_{q^n}$ and prove that $(\mathbb{Z}_{q^n}, d_q)$ is isometric to $(\mathbb{Z}_q^n, d_{RT})$ for every $n$, where $d_{RT}$ is the Rosenbloom-Tsfasman metric. More generally, we then demonstrate that any pair of finite groups of the same cardinality are isometric to each other for some metrics that can be explicitly constructed. Finally, we consider a chain $\mathcal{C}$ of subgroups of a given group and define the chain metric $d_{\mathcal{C}}$ and chain isometries between two chains. Let $G, K$ be groups with $|G|=q^n$, $|K|=q$ and let $H<G$. Using chains, we prove that under certain conditions, $(G,d_\mathcal{C}) \simeq (K^n, d_{RT})$ and $(G,d_\mathcal{C}) \simeq (H^{[G:H]}, d_{BRT})$ where $d_{BRT}$ is the block Rosenbloom-Tsfasman metric which generalizes $d_{RT}$.
For $q=p^m$ with $p$ prime and $k\mid q-1$, we consider the generalized Paley graph $Γ(k,q) = Cay(\mathbb{F}_q, R_k)$, with $R_k=\{ x^k : x \in \mathbb{F}_q^* \}$, and the irreducible $p$-ary cyclic code $\mathcal{C}(k,q) = \{(\textrm{Tr}_{q/p}(γω^{ik})_{i=0}^{n-1})\}_{γ\in \mathbb{F}_q}$, with $ω$ a primitive element of $\mathbb{F}_q$ and $n=\tfrac{q-1}{k}$. We first express the spectra of $Γ(k,q)$ in terms of Gaussian periods. Then, we show that the spectra of $Γ(k,q)$ and $\mathcal{C}(k,q)$ are mutually determined by each other if further $k\mid \tfrac{q-1}{p-1}$. We give $Spec(Γ(k,q))$ explicitly for those graphs associated with irreducible 2-weight cyclic codes in the semiprimitive and exceptional cases. We also compute $Spec(Γ(3,q))$ and $Spec(Γ(4,q))$.
We show geometrically that $\sqrt n$ is irrational for $n=3,5,7$ by adapting Tennenbaum's geometric proof that $\sqrt 2$ is irrational. We also show that this method cannot be used to prove the irrationality of $\sqrt n$ for a bigger $n$.
We consider cyclic codes $\mathcal{C}_\mathcal{L}$ associated to quadratic trace forms in $m$ variables $Q_R(x) = \operatorname{Tr}_{q^m/q}(xR(x))$ determined by a family $\mathcal{L}$ of $q$-linearized polynomials $R$ over $\mathbb{F}_{q^m}$, and three related codes $\mathcal{C}_{\mathcal{L},0}$, $\mathcal{C}_{\mathcal{L},1}$ and $\mathcal{C}_{\mathcal{L},2}$. We describe the spectra for all these codes when $\mathcal{L}$ is an even rank family, in terms of the distribution of ranks of the forms $Q_R$ in the family $\mathcal{L}$, and we also compute the complete weight enumerator for $\mathcal{C}_\mathcal{L}$. In particular, considering the family $\mathcal{L} = \langle x^{q^\ell} \rangle$, with $\ell$ fixed in $\mathbb{N}$, we give the weight distribution of four parametrized families of cyclic codes $\mathcal{C}_\ell$, $\mathcal{C}_{\ell,0}$, $\mathcal{C}_{\ell,1}$ and $\mathcal{C}_{\ell,2}$ over $\mathbb{F}_q$ with zeros $\{ α^{-(q^\ell+1)} \}$, $\{ 1,\, α^{-(q^\ell+1)} \}$, $\{ α^{-1},\,α^{-(q^\ell+1)} \}$ and $\{ 1,\,α^{-1},\,α^{-(q^\ell+1)}\}$ respectively, where $q = p^s$ with $p$ prime, $α$ is a generator of $\mathbb{F}_{q^m}^*$ and $m/(m,\ell)$ is even. Finally, we give simple necessary and sufficient conditions for Artin-Schreier curves $y^p-y = xR(x) + βx$, $p$ prime, associated to polynomials $R \in \mathcal{L}$ to be optimal. We then obtain several maximal and minimal such curves in the case $\mathcal{L} = \langle x^{p^\ell}\rangle$ and $\mathcal{L} = \langle x^{p^\ell}, x^{p^{3\ell}} \rangle$.
We study the asymptotic behavior of a family of algebraic geometry codes, which we call block-transitive, that generalizes the classes of transitive and quasi-transitive codes. We prove, by using towers of algebraic function fields, that there are sequences of codes in this family attaining the Tsfasman-Vladut-Zink bound over finite fields of square cardinality. We give the exact length of these codes as well as explicit lower bounds for their parameters.
We compute the eta function $η(s)$ and its corresponding $η$-invariant for the Atiyah-Patodi-Singer operator $\mathcal{D}$ acting on an orientable compact flat manifold of dimension $n =4h-1$, $h\ge 1$, and holonomy group $F\simeq \mathbb{Z}_{2^r}$, $r\in \mathbb{N}$. We show that $η(s)$ is a simple entire function times $L(s,χ_4)$, the $L$-function associated to the primitive Dirichlet character modulo 4. The $η$-invariant is 0 or equals $\pm 2^k$ for some $k\ge 0$ depending on $r$ and $n$. Furthermore, we construct an infinite family $\mathcal{F}$ of orientable $\mathbb{Z}_{2^r}$-manifolds with $F\subset \mathrm{SO}(n,\mathbb{Z})$. For the manifolds $M\in \mathcal{F}$ we have $η(M)=-\tfrac{1}{2}|T|$, where $T$ is the torsion subgroup of $H_1(M,\mathbb{Z})$, and that $η(M)$ determines the whole eta function $η(s,M)$.
We obtain new combinatorial identities for integral values of binary Krawtchouk polynomials $K^{2m}_p(x)$, $0\le p\le 2m$, by computing the characters of the $p$-exterior representations on certain elements of order 2 of $\mathrm{SO}(2m)$. From this identities, we deduce several new relations for binomial coefficients and Catalan numbers.