Talks

Tuesday, July 13, 12:00 ~ 12:25 UTC-3

The Generalized Covering Radii of Linear Codes

Marcelo Firer

Unicamp, Brazil   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak4f5ebc7c0b46727c8bf89f298b171328').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy4f5ebc7c0b46727c8bf89f298b171328 = 'mf&#105;r&#101;r' + '&#64;'; addy4f5ebc7c0b46727c8bf89f298b171328 = addy4f5ebc7c0b46727c8bf89f298b171328 + '&#117;n&#105;c&#97;mp' + '&#46;' + 'br'; var addy_text4f5ebc7c0b46727c8bf89f298b171328 = 'mf&#105;r&#101;r' + '&#64;' + '&#117;n&#105;c&#97;mp' + '&#46;' + 'br';document.getElementById('cloak4f5ebc7c0b46727c8bf89f298b171328').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy4f5ebc7c0b46727c8bf89f298b171328 + '\'>'+addy_text4f5ebc7c0b46727c8bf89f298b171328+'<\/a>';

In this work we generalize the concept of covering radius, introducing the $r$-covering radii of a linear code. The first part of the work present several equivalent definitions, that explores different aspects of the subject: combinatorial, geometric and algebraic notions. We describe the connection of the generalized covering radii with generalized Hamming weights and produce some asymptotic bounds that shows its relevance for application to database linear querying.

Joint work with Dor Elimelech and Moshe Schwartz (Ben Gurion University, Israel)..

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Tuesday, July 13, 12:30 ~ 12:55 UTC-3

Lattices from codes over finite rings and applications for reliable and secure communications

Sueli I. R. Costa

Lattices are discrete sets of the n-dimensional Euclidean space which are described by all integer linear combination of a set of independent vectors. Lattice coding have been proposed for signal transmission over AWGN and fading channels and also for cryptographic schemes in the so called post-quantum cryptography. We approach lattices constructed from linear codes over finite rings and applications. Construction A lattices with coding scheme through Voronoi constellations and multilevel lattice constructions $C^{*}$, $D$, $D’$ and $\bar{D}$ from families of nested linear codes are presented.

Joint work with Ana Paula de Souza ( University of Campinas, Brazil) and Eleonesio Strey (Federal University of Espirito Santo, Brazil).

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Tuesday, July 13, 13:00 ~ 13:25 UTC-3

The Curious Case of the Diamond Network

Allison Beemer

In this talk, we consider the one-shot capacity of communication networks subject to adversarial noise affecting a subset of network edges. In particular, we examine previously-established upper bounds on one-shot capacity. We introduce the Diamond Network as a minimal example to show that known cut-set bounds are not sharp in general, and give a capacity-achieving scheme for the Diamond Network that implements an adversary detection strategy. We conclude with more general results, including a family of networks for which known cut-set bounds are, in fact, tight.

Joint work with Alberto Ravagnani (Eindhoven University of Technology).

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Tuesday, July 13, 13:30 ~ 13:55 UTC-3

Twisted group codes

Javier de la Cruz

Universidad del Norte, Colombia   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakc2d2c441f8cb7984e2615fdd52a93abe').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyc2d2c441f8cb7984e2615fdd52a93abe = 'jd&#101;l&#97;cr&#117;z' + '&#64;'; addyc2d2c441f8cb7984e2615fdd52a93abe = addyc2d2c441f8cb7984e2615fdd52a93abe + '&#117;n&#105;n&#111;rt&#101;' + '&#46;' + '&#101;d&#117;' + '&#46;' + 'c&#111;'; var addy_textc2d2c441f8cb7984e2615fdd52a93abe = 'jd&#101;l&#97;cr&#117;z' + '&#64;' + '&#117;n&#105;n&#111;rt&#101;' + '&#46;' + '&#101;d&#117;' + '&#46;' + 'c&#111;';document.getElementById('cloakc2d2c441f8cb7984e2615fdd52a93abe').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyc2d2c441f8cb7984e2615fdd52a93abe + '\'>'+addy_textc2d2c441f8cb7984e2615fdd52a93abe+'<\/a>';

We investigate right ideals as codes in twisted group algebras. Such codes are called twisted group codes. It turns out that many interesting codes belong to this class; for instance, the ternary extended Golay code, Hamming codes and constacyclic codes. In particular we characterize all linear codes which are twisted group codes in terms of their automorphism group.

Joint work with Wolfgang Willems (Otto-von-Guericke-Universität, Magdeburg).

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Tuesday, July 13, 14:00 ~ 14:25 UTC-3

Additive twisted codes and new infinite families of binary quantum codes

Reza Dastbasteh

We study a subclass of additive cyclic codes called the additive twisted codes, which were introduced by Bierbrauer and Edel (1997). The twisted codes inherit many properties of linear cyclic codes such as the BCH minimum distance bound. We show that many other well-known minimum distance bounds for the linear cyclic codes including Hartman-Tzeng, Roos, and van Lint-Wilson minimum distance bounds all remain valid for the twisted codes.

A novel minimum distance condition for the twisted codes with a symmetric defining set is also provided. This result leads to new infinite classes of twisted codes with the minimum distance five. Following that, several new infinite families of good binary quantum codes with the minimum distances four and five are constructed.

J. Bierbrauer and Y. Edel: Quantum twisted codes, Journal of Combinatorial Designs 8 (2000), 174-188.

Y. Edel and J. Bierbrauer: Twisted BCH-codes, Journal of Combinatorial Designs 5 (1997), 377-389.

Joint work with Petr Lisonek (Simon Fraser University).

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Tuesday, July 13, 14:30 ~ 14:55 UTC-3

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

Denis Videla

Universidad Nacional de Córdoba, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak87bf89612b87a1576b05923d90ce001c').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy87bf89612b87a1576b05923d90ce001c = 'd&#101;n&#105;sv458' + '&#64;'; addy87bf89612b87a1576b05923d90ce001c = addy87bf89612b87a1576b05923d90ce001c + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m'; var addy_text87bf89612b87a1576b05923d90ce001c = 'd&#101;n&#105;sv458' + '&#64;' + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m';document.getElementById('cloak87bf89612b87a1576b05923d90ce001c').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy87bf89612b87a1576b05923d90ce001c + '\'>'+addy_text87bf89612b87a1576b05923d90ce001c+'<\/a>';

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. This talk is based on a recent published article.

\textsc{R.A.\@ Podest\'a, D.E.\@ Videla}. \textit{The weight distribution of irreducible cyclic codes associated with decomposable generalized Paley graphs}. Adv.\@ Math.\@ Comm.\@ (2021), doi: 10.3934/amc.2021002

Joint work with Ricardo Podestá (Universidad Nacional de Córdoba, Argentina).

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Tuesday, July 13, 15:00 ~ 15:25 UTC-3

On the number of solutions of systems of certain diagonal equations over finite fields.

Melina Privitelli

Let $\mathbb{F}_{\hskip-0.7mm q}$ be the finite field of $q$ elements. It is a classical problem to determine or to estimate the number $N$ of $\mathbb{F}_{\hskip-0.7mm q}$--rational solutions (i.e. solutions with coordinates in $\mathbb{F}_{\hskip-0.7mm q}$) of systems of polynomial equations over $\mathbb{F}_{\hskip-0.7mm q}$ (see, e.g., [5]). There are explicit formulas for the number $N$ only for some very particular cases (see, e.g., [1] and [8]) . For this reason, it is important to have estimates on the number $N$ and results which guarantee the existence of this kind of solutions.

Particularly, the systems of diagonal equations $$(1)\hspace{1in}\left \{\begin{array}{ccl} a_{11}X_1^{d_1} & + a_{12}X_2^{d_2} + \cdots + & a_{1t}X_t^{d_t} = b_1 \\ a_{21}X_1^{d_1} & + a_{22}X_2^{d_2} + \cdots + & a_{2t}X_t^{d_t} = b_2 \\ \;\vdots & &\quad\vdots \\ a_{n1}X_1^{d_1} &+ a_{n2}X_2^{d_2} + \cdots +& a_{nt}X_t^{d_t} = b_n, \end{array}\right.$$ with $b_1,\ldots,b_n\in \mathbb{F}_{\hskip-0.7mm q}$, have been considered in the literature because the study of its set of $\mathbb{F}_{\hskip-0.7mm q}$--rational solutions has several applications to different areas of mathematics, such as the theory of cyclotomy, Waring’s problem and the graph coloring problem (see, e.g. [3] and [5]). Additionally, information on the number $N$ is very useful in different aspects of coding theory such as the weight distribution of some cyclic codes ([9] and [10]) and the covering radius of certain cyclic codes ( [2] and [4]).

In comparison with a single diagonal equation, there are much fewer results about the number of $\mathbb{F}_{\hskip-0.7mm q}$-rational solutions of systems of the type (1) and most of them use tools involving character sums. In this work, we approach this problem using tools of algebraic geometry. More precisely, we consider an $\mathbb{F}_{\hskip-0.7mm q}$-variety $V$ associated to the system. We study the geometric properties of $V$, where the key point is obtaining upper bounds of the dimension of its singular locus. This study allows us to obtain estimates and existence results of rational solutions of systems of diagonal equations which in particular improve W. Spackman’s (see [6] and [7]). Furthermore, our techniques can be applied to the study of some variants of these systems such as systems of Dickson’s equations and generalized diagonal equations.

References.

[1] X. Cao, W-S. Chou and J. Gu, On the number of solutions of certain diagonal equations over finite fields, Finite Fields Appl. 42 (2016), 225--252.

[2] T. Helleseth, On the covering radius of cyclic linear codes and arithmetic codes, Discrete Appl. Math. 11(1985), no. 2, 157--173.

[3] R. Lidl and H. Niederreiter, Finite fields, Addison--Wesley, Reading, Massachusetts, 1983.

[4] O. Moreno and F. N. Castro, Divisibility properties for covering radius of certain cyclic codes, IEEE Trans. Inform. Theory 49 (2003), no. 12, 3299--3303.

[5] Gary L. Mullen and Daniel Panario, Handbook of Finite Fields (1st ed.), Chapman and Hall/CRC, 2013.

[6] K. W. Spackman, Simultaneous solutions to diagonal equations over finite fields, J. Number Theory 11(1979), no. 1, 100--115.

[7] K. W. Spackman, On the number and distribution of simultaneous solutions to diagonal congruences, Canadian J. Math. 33 (1981), no. 2, 421--436.

[8] J. Wolfmann, Some systems of diagonal equations over finite fields, Finite Fields Appl. 4(1998), no. 1, 29--37.

[9] X. Zeng L. Hu, W. Jiang, Q. Yue and X. Cao, The weight distribution of a class of $p$-ary cyclic codes, Finite Fields Appl. 16 (2010), no.1, 56--73.

[10] D. Zheng, X. Wang, X. Zeng and L. Hu, The weight distribution of a family of $p$-ary cyclic codes, Des. Codes Cryptogr. 75(2015), no. 2, 263--275.

Joint work with Mariana Pérez, Universidad Nacional de Hurlingham, CONICET.

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Tuesday, July 13, 15:30 ~ 15:55 UTC-3

Essential idempotents and Nilpotent Group Codes

César Polcino Milies

Universidade de São Paulo, Brazil   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakc147e4ced66f46e3b5d5e1939a29173e').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyc147e4ced66f46e3b5d5e1939a29173e = 'p&#111;lc&#105;n&#111;m&#105;l&#105;&#101;s' + '&#64;'; addyc147e4ced66f46e3b5d5e1939a29173e = addyc147e4ced66f46e3b5d5e1939a29173e + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m'; var addy_textc147e4ced66f46e3b5d5e1939a29173e = 'p&#111;lc&#105;n&#111;m&#105;l&#105;&#101;s' + '&#64;' + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m';document.getElementById('cloakc147e4ced66f46e3b5d5e1939a29173e').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyc147e4ced66f46e3b5d5e1939a29173e + '\'>'+addy_textc147e4ced66f46e3b5d5e1939a29173e+'<\/a>';

We recall the definition of essential idempotents and its implications for cyclic and Abelian codes. Then, we consider nilpotent group codes, i.e. codes that can be realized as ideals in the finite (semisimple) group algebra of a nilpotent group. We discuss the existence of essential idempotents in this context and study properties of minimal nilpotent codes.

References [1] G.Chalom, R. Ferraz and C.Polcino Milies, Essential idempotents and simplex codes, J. Algebra, Discrete Structures and Appl., 4, 2 (2017), 181-188. [2] G.Chalom, R. Ferraz and C.Polcino Milies, Essencial idempotents and codes of constant weight, Sâo Paulo J. of Math. Sci,, 11 (2) (2018), 253-260 [3] A. Duarte, On nilpotent and constacyclic codes, tese de doutoramento, Universidade Federal do ABC, Santo André, Brazil, 2021.

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Wednesday, July 14, 12:00 ~ 12:25 UTC-3

Sobre la dimensión de ideales en álgebras de grupo

Horacio Tapia-Recillas

Universidad Autónoma Metropolitáana-Iztapalapa, Mexico   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak469fe209cf10d7cc2896a4dae80446b4').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy469fe209cf10d7cc2896a4dae80446b4 = 'htr' + '&#64;'; addy469fe209cf10d7cc2896a4dae80446b4 = addy469fe209cf10d7cc2896a4dae80446b4 + 'x&#97;n&#117;m' + '&#46;' + '&#117;&#97;m' + '&#46;' + 'mx'; var addy_text469fe209cf10d7cc2896a4dae80446b4 = 'htr' + '&#64;' + 'x&#97;n&#117;m' + '&#46;' + '&#117;&#97;m' + '&#46;' + 'mx';document.getElementById('cloak469fe209cf10d7cc2896a4dae80446b4').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy469fe209cf10d7cc2896a4dae80446b4 + '\'>'+addy_text469fe209cf10d7cc2896a4dae80446b4+'<\/a>';

En esta plática se presentan cotas y relaciones para la dimensión de ideales principales en álgebras de grupo analizando el polinomio mínimo y característico de la representación regular asociada al generador del ideal. Estos resultados son usados en el contexto de álgebras de grupo semisimples para calcular la dimensión de códigos abelianos.

Joint work with Elias J. García-Claro (Universidad Autónoma Metropolitána-Iztapalapa, México).

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Wednesday, July 14, 12:30 ~ 12:55 UTC-3

Evaluation codes and their basic parameters

Delio Jaramillo-Velez

Cinvestav , Mexico   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakd1be20d5b34643501421032925b71a33').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyd1be20d5b34643501421032925b71a33 = 'dj&#97;r&#97;m&#105;ll&#111;' + '&#64;'; addyd1be20d5b34643501421032925b71a33 = addyd1be20d5b34643501421032925b71a33 + 'm&#97;th' + '&#46;' + 'c&#105;nv&#101;st&#97;v' + '&#46;' + 'mx'; var addy_textd1be20d5b34643501421032925b71a33 = 'dj&#97;r&#97;m&#105;ll&#111;' + '&#64;' + 'm&#97;th' + '&#46;' + 'c&#105;nv&#101;st&#97;v' + '&#46;' + 'mx';document.getElementById('cloakd1be20d5b34643501421032925b71a33').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyd1be20d5b34643501421032925b71a33 + '\'>'+addy_textd1be20d5b34643501421032925b71a33+'<\/a>';

The aim of this talk is to give degree formulas for the generalized Hamming weights of evaluation codes and to show lower bounds for these weights. In particular, we determine the minimum distance of toric codes over hypersimplices, and the 1st and 2nd generalized Hamming weights of squarefree evaluation codes.

Joint work with Maria Vaz Pinto (Instituto Superior Técnico, Universidade Técnica de Lisboa) and Rafael H. Villarreal (Cinvestav).

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Wednesday, July 14, 13:00 ~ 13:25 UTC-3

Algebraic interpretation of the minimum distance of Reed-Muller-type codes

Yuriko Pitones

The minimum distance of Reed-Muller-type codes has an algebraic interpretation, in terms of its associated vanishing ideal $I$ and the Hilbert-Samuel multiplicity of $I$, called the $\delta$-function. In this talk, we present this interpretation and study the asymptotic behavior of the $\delta$-function, in particular, we related the stabilization point, $r$ of $I$ of the $\delta$-function with the Castelnuovo-Mumford regularity of $I$. We see that when generalized the $\delta$-function to graded ideals, the point $r I$ is less than or equal to the Castelnuovo -Mumford regularity of $I$, in particular, this claim holds for F-pure and square free monomial ideals.

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Wednesday, July 14, 13:30 ~ 13:55 UTC-3

A family of codes with locality containing optimal codes

Victor Gonzalo Lopez Neumann

Universidade Federal de Uberlândia, Brazil   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakd95e4382621a32d81296045ed7f72037').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyd95e4382621a32d81296045ed7f72037 = 'v&#105;ct&#111;r.n&#101;&#117;m&#97;nn' + '&#64;'; addyd95e4382621a32d81296045ed7f72037 = addyd95e4382621a32d81296045ed7f72037 + '&#117;f&#117;' + '&#46;' + 'br'; var addy_textd95e4382621a32d81296045ed7f72037 = 'v&#105;ct&#111;r.n&#101;&#117;m&#97;nn' + '&#64;' + '&#117;f&#117;' + '&#46;' + 'br';document.getElementById('cloakd95e4382621a32d81296045ed7f72037').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyd95e4382621a32d81296045ed7f72037 + '\'>'+addy_textd95e4382621a32d81296045ed7f72037+'<\/a>';

Locally recoverable codes were introduced by Gopalan et al.\ in 2012, and in the same year, Prakash et al.\ introduced the concept of codes with locality, which are a type of locally recoverable codes. In this work, we introduce a new family of codes with locality, which are subcodes of a certain family of evaluation codes. We determine the dimension of these codes, and also bounds for the minimum distance. We present the true values of the minimum distance in special cases, and also show that elements of this family are optimal codes'', as defined by Prakash et al.

Joint work with Bruno Andrade (Universidade Federal de Uberlândia, Brazil) and Cícero Carvalho (Universidade Federal de Uberlândia, Brazil).

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Wednesday, July 14, 14:00 ~ 14:25 UTC-3

On the minimum distance of some codes parameterized by the edges of an even cycle

Manuel Gonzalez Sarabia

Several approaches have been given to the study of the codes parametrized by the edges of a graph. In this work we analyze the behavior of the minimum distance in the case of the codes parameterized by the edges of an even cycle. Although we know the main parameters in the case of odd cycles, including their complete weight hierarchy, because in this situation we are dealing with a projective torus, the case of even cycles remains unknown.

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Wednesday, July 14, 14:30 ~ 14:55 UTC-3

Decreasing monomial codes

Eduardo Camps-Moreno

Instituto Politécnico Nacional, México   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloaka4ace85277d1b6df8415132a03f3c105').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addya4ace85277d1b6df8415132a03f3c105 = '&#101;cfmd' + '&#64;'; addya4ace85277d1b6df8415132a03f3c105 = addya4ace85277d1b6df8415132a03f3c105 + 'h&#111;tm&#97;&#105;l' + '&#46;' + 'c&#111;m'; var addy_texta4ace85277d1b6df8415132a03f3c105 = '&#101;cfmd' + '&#64;' + 'h&#111;tm&#97;&#105;l' + '&#46;' + 'c&#111;m';document.getElementById('cloaka4ace85277d1b6df8415132a03f3c105').innerHTML += '<a ' + path + '\'' + prefix + ':' + addya4ace85277d1b6df8415132a03f3c105 + '\'>'+addy_texta4ace85277d1b6df8415132a03f3c105+'<\/a>';

We describe the class of decreasing monomial codes, which are evaluation codes that generalizes the well-known families of Reed-Muller codes, Affine Cartesian Codes, Hyperbolic codes and more. We describe its parameters and explain how they are related with polar codes.

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Wednesday, July 14, 15:00 ~ 15:25 UTC-3

Hermitian-Lifted Codes

Beth Malmskog

Colorado College, United States   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak68f3ed302b7fc5a564f423e367ae0da4').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy68f3ed302b7fc5a564f423e367ae0da4 = 'bm&#97;lmsk&#111;g' + '&#64;'; addy68f3ed302b7fc5a564f423e367ae0da4 = addy68f3ed302b7fc5a564f423e367ae0da4 + 'c&#111;l&#111;r&#97;d&#111;c&#111;ll&#101;g&#101;' + '&#46;' + '&#101;d&#117;'; var addy_text68f3ed302b7fc5a564f423e367ae0da4 = 'bm&#97;lmsk&#111;g' + '&#64;' + 'c&#111;l&#111;r&#97;d&#111;c&#111;ll&#101;g&#101;' + '&#46;' + '&#101;d&#117;';document.getElementById('cloak68f3ed302b7fc5a564f423e367ae0da4').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy68f3ed302b7fc5a564f423e367ae0da4 + '\'>'+addy_text68f3ed302b7fc5a564f423e367ae0da4+'<\/a>';

In recent work of Lopez, Malmskog, Matthews, Pinero-Gonzales, and Wootters, we constructed codes for local recovery of erasures with high availability and constant-bounded rate from the Hermitian curve. These new codes, called Hermitian-lifted codes, are evaluation codes with evaluation set being the set of $F_(q^2)$-rational points on the affine curve. The novelty is in terms of the functions to be evaluated; they are a special set of monomials which restrict to low degree polynomials on lines intersected with the Hermitian curve. The resulting codes are neither punctured traditional lifted codes, nor subcodes of previously defined locally recoverable codes on the Hermitian curve. This talk will introduce the codes and bounds on their parameters, and discuss questions for further research.

Joint work with Hiram Lopez, Valdez (University of Cleveland), Gretchen Matthews (Virginia Tech), Fernando Pinero-Gonzales (University of Puerto Rico Ponce) and Mary Wootters (Stanford University).

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Wednesday, July 14, 15:30 ~ 15:55 UTC-3

Toric 3-fold codes and Minkowski length of lattice polytopes in $\mathbb{R}^3$

Jenya Soprunova

Fix a convex lattice polytope $P$ in $\mathbb{R}^n$, and define ${\mathcal L}_P$ to be the $\mathbb{F}_q$-vector space spanned by the monomials whose exponent vectors lie in $P$. The codewords of a toric code are obtained by evaluating polynomials in ${\mathcal L}_P$ at the points of the torus $(\mathbb{F}_q\setminus\{0\})^n$, taken in some fixed order. The question of computing or giving bounds on the minimum distance of toric codes has been studied by Hansen, Joyner, Little and Schenck, and others.

Our goal is to provide an estimate on the minimum distance of a toric code in terms of geometric invariants of $P$. In our earlier work, focusing on the case $n=2$, we provided such estimates involving a geometric invariant $L(P)$, the Minkowski length of $P$. In this talk I will concentrate on the case $n=3$ which involves much more challenging combinatorics as well as phenomena that do not arise in the case of toric surface codes.

Joint work with Kyle Meyer (UC San Diego) and Ivan Soprunov (Cleveland State University).

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Monday, July 19, 16:00 ~ 16:25 UTC-3

Structure and Parameters of Sum-Rank-Metric Codes

Alberto Ravagnani

Eindhoven University of Technology, the Netherlands   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak550bf11ccc5a703598ecd132b88493f1').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy550bf11ccc5a703598ecd132b88493f1 = '&#97;.r&#97;v&#97;gn&#97;n&#105;' + '&#64;'; addy550bf11ccc5a703598ecd132b88493f1 = addy550bf11ccc5a703598ecd132b88493f1 + 't&#117;&#101;' + '&#46;' + 'nl'; var addy_text550bf11ccc5a703598ecd132b88493f1 = '&#97;.r&#97;v&#97;gn&#97;n&#105;' + '&#64;' + 't&#117;&#101;' + '&#46;' + 'nl';document.getElementById('cloak550bf11ccc5a703598ecd132b88493f1').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy550bf11ccc5a703598ecd132b88493f1 + '\'>'+addy_text550bf11ccc5a703598ecd132b88493f1+'<\/a>';

I will discuss the fundamental properties of error-correcting codes endowed with the sum-rank metric. These are linear spaces of tuple of matrices (of mixed size) endowed with the sum-rank distance. The new results I will present include upper bounds for the size of sum-rank-metric codes, existence results, and duality properties.

Joint work with Eimear Byrne (University College Dublin) and Heide Gluesing-Luerssen (University of Kentucky).

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Monday, July 19, 16:30 ~ 16:55 UTC-3

Interference Alignment in Multiple Unicast Networks over Finite Fields

Felcie Manganiello

Clemson University, United States of America   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak972b0f4923589b414320e91294537e8d').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy972b0f4923589b414320e91294537e8d = 'm&#97;ng&#97;nm' + '&#64;'; addy972b0f4923589b414320e91294537e8d = addy972b0f4923589b414320e91294537e8d + 'cl&#101;ms&#111;n' + '&#46;' + '&#101;d&#117;'; var addy_text972b0f4923589b414320e91294537e8d = 'm&#97;ng&#97;nm' + '&#64;' + 'cl&#101;ms&#111;n' + '&#46;' + '&#101;d&#117;';document.getElementById('cloak972b0f4923589b414320e91294537e8d').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy972b0f4923589b414320e91294537e8d + '\'>'+addy_text972b0f4923589b414320e91294537e8d+'<\/a>';

The origin of communication is based on the concept of two users exchanging information with each other over a single channel. The problem of perfect communication over a channel was modeled by Shannon in the late 40s. More modern communication networks are not so restrictive though. Most of the networks we use nowadays, connect multiple parties and graphs can be exploited to represent these networks. The question we are going to investigate in this seminar is simple: given a graph representing a network, what is its capacity, meaning how much information can be sent through it, and by which communication protocol? This question has been already answered for unicast networks, meaning networks between a single source and a single receiver, and for multicast networks, meaning networks used by a source to communicate simultaneously to multiple receivers. The capacity of communication for most networks with multiple sources is still an open question. Networks of this type are characterized by interference that is represented by the messages sent by undesired sources. A communication strategy has to be determined in order to remove the interference. We will focus our work on multiple unicast networks and look at the effectiveness of a practice known as interference alignment. We will define the concepts of achievable rate regions of a network and discover that the points of this region are in relation with what we define to be unambiguous codes. Finally, we will give some preliminary bounds.

Joint work with Frank Kschischang (University of Toronto, Canada), Alberto Ravagnani (Eindhoven University of Technology, Netherlands) and Kristen Savary (Clemson University, USA).

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Monday, July 19, 17:00 ~ 17:25 UTC-3

Polarization and channel memory

Stephen Timmel

Virginia Tech, United States   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak91b6669645ddfd5cb9cd143981cf02cf').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy91b6669645ddfd5cb9cd143981cf02cf = 'st&#105;mm&#101;l' + '&#64;'; addy91b6669645ddfd5cb9cd143981cf02cf = addy91b6669645ddfd5cb9cd143981cf02cf + 'vt' + '&#46;' + '&#101;d&#117;'; var addy_text91b6669645ddfd5cb9cd143981cf02cf = 'st&#105;mm&#101;l' + '&#64;' + 'vt' + '&#46;' + '&#101;d&#117;';document.getElementById('cloak91b6669645ddfd5cb9cd143981cf02cf').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy91b6669645ddfd5cb9cd143981cf02cf + '\'>'+addy_text91b6669645ddfd5cb9cd143981cf02cf+'<\/a>';

In 2009, Arikan developed a polarization algorithm which produces codes achieving the symmetric capacity of a binary-input discrete memoryless channel W. Since then, polarization has been applied to a variety of other channel models. However, comparatively little research has focused on the possibility that communication channels might depend on each other and on nearby symbols in the codeword. We present theorems and examples which further clarify when processes with channel memory polarize and what asymptotic rates can be expected.

Joint work with Gretchen Matthews (Virginia Tech, United States).

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Monday, July 19, 17:30 ~ 17:55 UTC-3

On Weierstrass gaps at several points

Guilherme Tizziotti

Universidade Federal de Uberlândia, Brazil   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak90402613471f41fd93da7c9a7c9c7f62').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy90402613471f41fd93da7c9a7c9c7f62 = 'g&#117;&#105;lh&#101;rm&#101;ct' + '&#64;'; addy90402613471f41fd93da7c9a7c9c7f62 = addy90402613471f41fd93da7c9a7c9c7f62 + '&#117;f&#117;' + '&#46;' + 'br'; var addy_text90402613471f41fd93da7c9a7c9c7f62 = 'g&#117;&#105;lh&#101;rm&#101;ct' + '&#64;' + '&#117;f&#117;' + '&#46;' + 'br';document.getElementById('cloak90402613471f41fd93da7c9a7c9c7f62').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy90402613471f41fd93da7c9a7c9c7f62 + '\'>'+addy_text90402613471f41fd93da7c9a7c9c7f62+'<\/a>';

In this talk we will consider the problem of determining Weierstrass gaps and pure Weierstrass gaps at several points. Using the notion of relative maximality in generalized Weierstrass semigroups due to Delgado, we present a description of these elements which generalizes the approach of Homma and Kim [1] given for pairs. Through this description, we present a study of the gaps and pure gaps at several points on a certain family of curves with separated variables.

Main References

[1] M. Homma and S. J. Kim, Goppa codes with Weierstrass pairs, J. Pure Appl. Algebra 162, 2001 , 273-290.

[2] W. Tenório and G. Tizziotti, Generalized Weierstrass semigroups and Riemann-Roch spaces for certain curves with separated variables, Finite Fields and Their Applications 57, 2019, 230-248.

[3] W. Tenório and G. Tizziotti, On Weierstrass gaps at several points, Bull. Brazilian Math. Society, 50, 2019, 543-559.

Joint work with Wanderson Tenório (Universidade Federal de Goiás).

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Monday, July 19, 18:00 ~ 18:25 UTC-3

Isometry-Dual Property in Flags of Two-Point AG codes

Alonso Sepúlveda Castellanos

University Federal of Uberlândia, Brazil   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakb7330fc89d8afa03d33dfb93090251ed').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyb7330fc89d8afa03d33dfb93090251ed = '&#97;l&#111;ns&#111;.c&#97;st&#101;ll&#97;n&#111;s' + '&#64;'; addyb7330fc89d8afa03d33dfb93090251ed = addyb7330fc89d8afa03d33dfb93090251ed + '&#117;f&#117;' + '&#46;' + 'br'; var addy_textb7330fc89d8afa03d33dfb93090251ed = '&#97;l&#111;ns&#111;.c&#97;st&#101;ll&#97;n&#111;s' + '&#64;' + '&#117;f&#117;' + '&#46;' + 'br';document.getElementById('cloakb7330fc89d8afa03d33dfb93090251ed').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyb7330fc89d8afa03d33dfb93090251ed + '\'>'+addy_textb7330fc89d8afa03d33dfb93090251ed+'<\/a>';

For $P$ and $Q$ rational places in a function field $\mathcal{F}$, we investigate the existence of isometry-dual flags of codes in the families of two-point algebraic geometry codes $$C_\mathcal L(D, a_0P+bQ)\subsetneq C_\mathcal L(D, a_1P+bQ)\subsetneq \dots \subsetneq C_\mathcal L(D, a_sP+bQ),$$ where the divisor $D$ is the sum of pairwise different rational places of $\mathcal{F}$ and $P, Q$ are not in $supp(D)$. We characterize those sequences in terms of $b$ for general function fields.

We then apply the result to the broad class of Kummer extensions $\mathcal{F}$ defined by affine equations of the form $y^m=f(x)$, for $f(x)$ a separable polynomial of degree $r$, where $\gcd(r, m)=1$. For $P$ the rational place at infinity and $Q$ the rational place associated to one of the roots of $f(x)$, it is shown that the flag of two-point algebraic geometry codes has the isometry-dual property if and only if $m$ divides $2b+1$. At the end we illustrate our results by applying them to two-point codes over several well know function fields.

Joint work with Luciane Quoos (University Federal of Rio de Janeiro, Brazil) and Maria Bras-Amorós (Universitat Rovira i Virgili, Spain).

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Monday, July 19, 18:30 ~ 18:55 UTC-3

Weierstrass pure gaps and codes on curves with three distinguished points

Herivelto Borges

In this talk we consider the class of smooth plane curves of degree $n+1>3$ over a finite field containing three points, $P_1,P_2,$ and $P_3$, such that $nP_1+P_2$, $nP_2+P_3$, and $nP_3+P_1$ are divisors cut out by three distinct lines. For any such a curve, the dimension of certain special divisors supported on $\{P_1,P_2,P_3\}$ is computed, and an explicit description of the set of all pure gaps at any subset of $\{P_1,P_2,P_3\}$ is provided. From this class of curves, which includes the Hermitian curve, one can construct Goppa codes having minimum distance better than the Goppa bound.

Joint work with Gregory Cunha (Universidade Federal de Goias).

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Monday, July 19, 19:00 ~ 19:25 UTC-3

$t$-graph of distances of a finitely-generated group and block codes

Ismael Gutierrez

Universidad del Norte, Colombia   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak98069e127402e3298086f5d60e12c19c').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy98069e127402e3298086f5d60e12c19c = '&#105;sg&#117;t&#105;&#101;r' + '&#64;'; addy98069e127402e3298086f5d60e12c19c = addy98069e127402e3298086f5d60e12c19c + '&#117;n&#105;n&#111;rt&#101;' + '&#46;' + '&#101;d&#117;' + '&#46;' + 'c&#111;'; var addy_text98069e127402e3298086f5d60e12c19c = '&#105;sg&#117;t&#105;&#101;r' + '&#64;' + '&#117;n&#105;n&#111;rt&#101;' + '&#46;' + '&#101;d&#117;' + '&#46;' + 'c&#111;';document.getElementById('cloak98069e127402e3298086f5d60e12c19c').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy98069e127402e3298086f5d60e12c19c + '\'>'+addy_text98069e127402e3298086f5d60e12c19c+'<\/a>';

Let $G$ be a finitely-generated group with generating set $M=\{g_1,\ldots, g_n\}$, and suppose that every element in $x\in G$ can be uniquely written as $x=\prod_{i=1}^n g_i^{\epsilon_i}$. The $t$-graph of distances of $G$ is defined as the graph with vertices set $G$, and in which two vertices $x=\prod_{i=1}^n g_i^{\epsilon_i}$ and $y=\prod_{i=1}^n g_i^{\delta_i}$ are adjacent if the Minkowski distance between them is equal to $t$. That is, $l_1(x,y) = \sum_{i=1}^n |\epsilon_i-\delta_i| =t$. If $\mathscr{C}$ is subgroup of $G$, then we say that $\mathscr{C}$ is a group code. In this talk we consider such codes and the connection with $t$-graph of distances of $G$, when $t=1$.

Joint work with Elias Claro, UAM, CDMX.

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Monday, July 19, 19:30 ~ 19:55 UTC-3