Weight distribution of cyclic codes defined by quadratic forms and related curves

We consider cyclic codes CL associated to quadratic trace forms inm variables (Formula Presented) determined by a family L of q-linearized polynomials R over Fqm, and three related codes CL,0, CL,1, and CL,2. We describe the spectra for all these codes when L is an even rank family, in terms of the distribution of ranks of the forms QR in the family L, and we also computethe complete weight enumerator for CL. In particular, considering the family L = ‹xql›, with l fixed in N, we give the weight distribution of four parametrized families of cyclic codes Cl, Cl,0,Cl,1, and Cl,2 over Fq with zeros(Formula Presented) respectively,where q = ps with p prime, α is a generator of F*qm, and m/(m,l)is even. Finally, we give simple necessary and sufficient conditions for Artin–Schreier curves yp−y = xR(x)+βx, p prime, associated to polynomials R ∈ L to be optimal. We then obtain several maximal and minimal such curves inthe case (Formula Presented).Fil: Podesta, Ricardo Alberto. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; ArgentinaFil: Videla Guzman, Denis Eduardo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentin

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

We use known characterizations of generalized Paley graphs which areCartesian decomposable to explicitly compute the spectra of the corresponding associated irreducible cyclic codes. As applications, we give reduction formulas forthe number of rational points in Artin-Schreier curves defined over extension fields and to the computation of Gaussian periods.Fil: Podesta, Ricardo Alberto. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; ArgentinaFil: Videla Guzman, Denis Eduardo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentin

The conorm code of an AG-code

Given a suitable extension F′/F of algebraic function fields over a finite field Fq, we introduce the conorm code ConF′/F(C) defined over F′ which is constructed from an algebraic geometry code C defined over F. We study the parameters of ConF′/F(C) in terms of the parameters of C, the ramification behavior of the places used to define C and the genus of F. In the case of unramified extensions of function fields we prove that ConF′/F(C)⊥=ConF′/F(C⊥) when the degree of the extension is coprime to the characteristic of Fq. 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.Fil: Chara, María de Los Ángeles. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional del Litoral. Facultad de Ingeniería Química; ArgentinaFil: Podesta, Ricardo Alberto. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; ArgentinaFil: Toledano, Ricardo Daniel. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional del Litoral. Facultad de Ingeniería Química; Argentin

MASTER Optical Polarization Variability Detection in the Microquasar V404 Cyg/GS 2023+33

On 2015 June 15, the Swift space observatory discovered that the Galactic black hole candidate V404 Cyg was undergoing another active X-ray phase, after 25 years of inactivity. The 12 telescopes of the MASTER Global Robotic Net located at six sites across four continents were the first ground-based observatories to start optical monitoring of the microquasar after its gamma-ray wake up at 18h 34m 09s U.T. on 2015 June 15. In this paper, we report, for the first time, the discovery of variable optical linear polarization, changing by 4%-6% over a timescale of ∼1 hr, on two different epochs. We can conclude that the additional variable polarization arises from the relativistic jet generated by the black hole in V404 Cyg. The polarization variability correlates with optical brightness changes, increasing when the flux decreases.Fil: Lipunov, V.. M.V.Lomonosov Moscow State University. Physics Department; RusiaFil: Gorbovskoy, E.. M.V.Lomonosov Moscow State University, Sternberg Astronomical Institute; RusiaFil: Krushinskiy, V.. Kourovka Astronomical Observatory, Ural Federal University; RusiaFil: Vlasenko, D.. M.V.Lomonosov Moscow State University, Sternberg Astronomical Institute; RusiaFil: Tiurina, N.. M.V.Lomonosov Moscow State University, Sternberg Astronomical Institute; RusiaFil: Balanutsa, P.. M.V.Lomonosov Moscow State University, Sternberg Astronomical Institute; RusiaFil: Kuznetsov, A.. M.V.Lomonosov Moscow State University, Sternberg Astronomical Institute; RusiaFil: Budnev, N.. Applied Physics Institute. Irkutsk State University; RusiaFil: Gress, O.. Applied Physics Institute, Irkutsk State University; RusiaFil: Tlatov, A.. Kislovodsk Solar Station of the Main (Pulkovo) Observatory RAS; RusiaFil: Rebolo Lopez, L.. Instituto de Astrofsica de Canarias; EspañaFil: Serra-Ricart, M.. Instituto de Astrofsica de Canarias; EspañaFil: Buckley, D. A. H.. South African Astronomical Observatory; SudáfricaFil: Israelyan, G.. Instituto de Astrofsica de Canarias; EspañaFil: Lodieu, N.. Instituto de Astrofisica de Canarias; EspañaFil: Ivanov, K.. Applied Physics Institute. Irkutsk State University; RusiaFil: Yazev, S.. Applied Physics Institute, Irkutsk State University; RusiaFil: Sergienko, Y.. Blagoveschensk State Pedagogical University; RusiaFil: Gabovich, A.. Blagoveschensk State Pedagogical University; RusiaFil: Yurkov, V.. Blagoveschensk State Pedagogical University; RusiaFil: Levato, Orlando Hugo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - San Juan. Instituto de Ciencias Astronómicas, de la Tierra y del Espacio. Universidad Nacional de San Juan. Instituto de Ciencias Astronómicas, de la Tierra y del Espacio; ArgentinaFil: Saffe, Carlos. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - San Juan. Instituto de Ciencias Astronómicas, de la Tierra y del Espacio. Universidad Nacional de San Juan. Instituto de Ciencias Astronómicas, de la Tierra y del Espacio; ArgentinaFil: Podesta, R.. Observatorio "Felix Aguiklar". Universidad Nacional de San Juan; ArgentinaFil: Lopez, C.. Observatorio "Felix Aguilar". Universidad nacional de San juan; Argentin

The eta function and eta invariant of Z2r -manifolds

We compute the eta function #x03B7;(s) and its corresponding η-invariant for the Atiyah–Patodi–Singer operator D acting on an orientable compact flat manifold of dimension =4h−1, ≥1, and holonomy group F≃Z2r , r∈N. We show that η(s) is a simple entire function times L(s,χ4), the L-function associated to the primitive Dirichlet character modulo 4. The η-invariant is 0 or equals ±2k for some k≥0 depending on r and n. Furthermore, we construct an infinite family F of orientable Z2r -manifolds with F⊂SO(n,Z). For the manifolds M∈F we have η(M)=−|T|, where T is the torsion subgroup of H1(M,Z), and that η(M) determines the whole eta function η(s,M).Fil: Podesta, Ricardo Alberto. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentin

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.Fil: Podesta, Ricardo Alberto. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; ArgentinaFil: Vides, Maximiliano Guillermo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentin

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 (Formula presented.). 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 ((Formula presented.) or (Formula presented.)). Then we show that, up to complements, the only bipartite regular graphs equienergetic and non-isospectral with their complements are the crown graphs (Formula presented.) or (Formula presented.). Next, for the family of strongly regular graphs Γ we characterize all possible parameters (Formula presented.) such that (Formula presented.). 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 (Formula presented.), (Formula presented.) and (Formula presented.) in Cameron's hierarchy (the cases (Formula presented.) in the non-bipartite case and (Formula presented.) are still open). Finally, we consider unitary Cayley graphs over rings (Formula presented.). We show that if R is a finite Artinian ring with an even number of local factors, then (Formula presented.) is complementary equienergetic if and only if (Formula presented.) is the product of 2 finite fields.Fil: Podesta, Ricardo Alberto. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; ArgentinaFil: Videla Guzman, Denis Eduardo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentin

Integral equienergetic non-isospectral unitary Cayley graphs

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 GR=X(R,R⁎), where R is a finite commutative ring with identity. We show that under mild conditions, {GR,GR+} are pairs of integral equienergetic non-isospectral graphs (generically connected and non-bipartite). Then, we obtain conditions such that {GR,G¯R} are equienergetic non-isospectral graphs. Finally, we characterize all integral equienergetic non-isospectral triples {GR,GR+,G¯R} such that all the graphs are also Ramanujan.Fil: Podesta, Ricardo Alberto. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; ArgentinaFil: Videla Guzman, Denis Eduardo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentin

Waring numbers over finite commutative local rings

In this paper we study Waring numbers gR(k) for (R,m) a finite commutative local ring with identity and k∈N with (k,|R|)=1. We first relate the Waring number gR(k) with the diameter of the Cayley graphs GR(k)=Cay(R,UR(k)) and WR(k)=Cay(R,SR(k)) with UR(k)={xk:x∈R⁎} and SR(k)={xk:x∈R×}, distinguishing the cases where the graphs are directed or undirected. We show that in both cases (directed or undirected), the graph GR(k) can be obtained by blowing-up the vertices of GFq (k) a number |m| of times, with independence sets the cosets of m, where q is the size of the residue field R/m. Then, by using the above blowing-up, we reduce the study of the Waring number gR(k) over the local ring R to the computation of the Waring number g(k,q) over the finite residue field R/m≃Fq. 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.Fil: Podesta, Ricardo Alberto. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; ArgentinaFil: Videla Guzman, Denis Eduardo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentin