# Program

The first Latin American Week on Coding and Information consists of a three-day school followed by a three-day workshop.
Students can apply for the whole week program and researchers for the workshop.
Our aim is to bring together researchers in all aspects of coding and information theory from all over the world, with special emphasis on Latin American researchers. The program aims to be attended by mathematicians, engineers and computer scientists. The LAWCI is an official satellite event of the International Congress of Mathematicians 2018, to be held in Rio de Janeiro in the following week.

# School

Sunday (July 22nd) | Monday (July, 23rd) | Tuesday (July, 24th) | |
---|---|---|---|

9:00 - 10:50 | Daniel Panario 1 | Daniel Panario 2 | Max Costa 2 |

10:50 - 11:10 | Coffee | Coffee | Coffee |

11:10 - 12:20 | Max Costa 1 | Posters - slides presentations | Ram Zamir |

12:20 - 13:00 | Max Costa 1 | Lunch | Lunch |

13:00 - 13:40 | Lunch (Confraternization) | Lunch | Lunch |

13:40 - 15:30 | Lunch (Confraternization) | Markus Grassl 1 | Markus Grassl 2 |

15:30 - 16:10 | Coffee + Poster Session | Coffee + Poster Session | |

16:10 - 18:00 | Moshe Schwarz 1 | Moshe Schwarz 2 |

# Workshop

**Oral presentations will be 20 minutes long, including up to 5 minutes for questions.

__JULY 22-24 - INTENSIVE SCHOOL FOR GRADUATE STUDENTS__Support kindly provided by IEEE - Information Theory Society

__CONFERENCE SUBJECTS__• coding theory: error-correcting codes, decoding algorithms, related combinatorial problems;

• discrete mathematics and algorithmic tools related to these two areas including: finite fields, discrete geometry, lattices and related algebraic structures.

**CONFIRMED LECTURES****Daniel Panario (Carleton University, Canada)**

**Title:**Finite fields, applications and open problems

**Abstract:**After a brief revision of finite fields concepts we describe several applications of finite fields in combinatorics. These include Latin squares and several types of combinatorial arrays and some of their applications and conexions to coding theory and cryptography. In the second lecture we review finite fields applications to cryptography, focusing on the differential map, permutation polynomials. It time allows we also comment on how random a pseudorandom sequence is, and criteria for cryptographically good pseudorandom sequences. We provide open problems for each of these topics.

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**Markus Grassl (Max Planck Institute, Erlangen, Germany)**

**Title:**Quantum Error-Correcting Codes: Discrete Maths meets Physics

**Abstract:**Quantum error-correcting codes (QECCs) are essential for building a quantum computer. Without assuming prior knowledge in quantum mechanics, the basic principles of QECCs will be introduced. This includes both the modeling of quantum information and quantum channels as well as the construction of QECCs based on classical error-correcting codes. It will be shown how the discrete structure enables efficient algorithms for encoding QECCs. Special attention will be given to CSS codes and stabilizer codes. In the second part, some more specific concepts will be highlighted, including code concatenation and quantum MDS codes.

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**Max Costa (Unicamp, Brazil)**

**Title:**Information Theory Fundamentals and Multiple User Applications

**Abstract:**This seminar will review the fundamental concepts of information theory and focus on applications in communications systems with multiple users. We will highlight the additive white Gaussian noise cases and show that the search for capacity has an interesting water filling interpretation. Multiple access, broadcast, interference and relay channels will be considered as examples.

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**Moshe Schwartz (Ben-Gurion University, Israel)**

**Title:**Coding for DNA Storage in Live Organisms

**Abstract:**DNA storage is slowly becoming a reality, bringing the promise of a super-dense storage medium. Recent experiments show it is possible to store information in living organisms, perhaps for unbounded time, as we ourselves are evidence of. Obviously, any storage system is susceptible to errors, and in the case of living DNA, mutations.

The coding literature has extensive knowledge on substitution errors, and insertions/deletion, as these are common in other application. However, a rarely studied but common form of mutation is duplication, causing repetitions in DNA sequences. Amazingly, the majority of the human genome is made up of repeated sequences. Repetitions were shown to be connected with diseases such as as cancer, myotonic dystrophy, Huntington's disease, and important phenomena such as chromosome fragility, expansion diseases, silencing genes, and rapid morphological variation. Repetitions are common in other species as well, and are claimed to be a major evolutionary force during vertebrate evolution.

In this talk we mathematically model string duplication, and ask several coding-theoretic questions:

1. Is there new information created strictly by duplication? What is the capacity of such systems? Is there a difference between the combinatorial (adversarial) model and the probabilistic model?

2. Can string duplication account for diversity? Can we reach every possible substring?

3. How do we correct strings corrupted by duplication?

4. What happens when we mix string-duplication errors with point mutations?

The talk is based on joint works with Ohad Elishco, Farzad Farnoud, Siddharth Jain, Yonatan Yehezkeally, and Jehoshua Bruck.

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**Ram Zamir (Tel Aviv University, Israel) - ITSoc Distinguished Lecturer (TBC)**

**Title:**On the Superiority of Equiangular Tight Frames

**Abstract:**Over-complete bases (frames) can be thought of as analog codes for compression (compressed sensing) or for redundant signaling (noise and erasure correction). Their goodness can be determined by the eigenvalue distribution of a random subset of the frame vectors. We focus on the highly symmetric class of equiangular tight frames (ETF). We show first that if we keep the frame aspect ratio and the selection probability fixed, then the eigenvalue distribution of a random subset of ETF converges to the MANOVA distribution in the limit as the dimension goes to infinity. Secondly, random subsets of ETF minimize the first four moments among all frames of a given aspect ratio. Finally, we conjecture that ETFs minimize also the "inverse moment", i.e., the noise enhancement under subset inversion. If true, then ETFs are the most robust analog codes, and their noise enhancement is asymptotically given by the inverse moment of the MANOVA distribution.

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**JULY 25-27 - WORKSHOP**Call for participation

**INVITED SPEAKERS****Alexander Barg (University of Maryland, US)**

**Title:**Different facets of the repair problem

**Abstract:**The repair problem refers to correcting one or several erasures with a given error-correcting code using as little inter-nodal communication as possible. An information-theoretic lower bound on the repair bandwidth is known from the literature, and codes that meet it with equality are said to support optimal repair. In this talk we consider several versions of the repair problem, illustrating different techniques behind the construction of optimal codes. We begin with describing an approach to optimal-repair codes using the notion of interference alignment, presenting several families of codes with a number of additional properties (universality, error tolerance, optimal updates). Then we discuss optimal repair of Reed-Solomon codes, presenting codes from this family that support optimal repair of a single failed node or of several failed nodes. In the final part of the talk we discuss one more setting of the problem, where the task is to perform repair of several nodes in a distributed way, and again construct a family of codes with optimal repair bandwidth. Based on joint works with Min Ye and Itzhak Tamo.

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**Gadiel Seroussi (Universidad de La República, Uruguay)**

**Title:**Q-ary Antipodal Matchings and Applications

**Abstract:**We study the problem of encoding arbitrary data into n x n arrays over the integer alphabet Q={0,1, ... , q-1}, all of whose row and column sums are at most n(q-1)/2. For this purpose, we define a q-ary antipodal matching to be a perfect matching in the bipartite graph with vertices corresponding to words of length m over Q, with left vertices corresponding to words with component sums greater m(q-1)/2, right vertices corresponding to words with component sums smaller than m(q-1)/2, and wherein two vertices are connected by an edge if one of the corresponding words dominates the other. We present two different constructions of efficiently computable q-ary antipodal matchings. We then show how such matchings can be used to implement the mentioned encodings into n x n q-ary matrices. The problem is motivated by the need to mitigate parasitic currents in envisioned multi-level memory systems based on crossbar arrays of resistive devices.

(Based on joint work with Erik Ordentlich and Ronny Roth.)

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**Olgica Milenkovic (ECE Illinois, US)**

**Title:**String reconstruction problems inspired by problems in -omic data analysis

**Abstract:**String reconstruction problems frequently arise in many areas of genomic data processing and synthetic biology. In the most general setting, they may be described as follows: one is given a single or multiple copies of a coded or uncoded string, and the copies are subsequently subjected to some form of (random) processing such as fragmentation or repeated transmission through a noise-inducing channel. The goal of the reconstruction method is to obtain an exact or approximate version of the string based on the processed outputs. Examples of string reconstruction questions include reconstruction from noisy traces, reconstruction from substrings and k-decks and reconstruction from compositional substring information. We review the above and some related problems and then proceed to describe coding methods that lead to strings that can be reconstructed exactly from their noisy traces and substrings. In particular, we focus on DNA profile codes, hybrid reconstruction from traces and uniquely reconstructable code designs. In the process, we introduce new questions in the areas of restricted de Bruin graphs, counting of rational points in polytopes and string replacement methods.

(This is a joint work with Ryan Gabrys, Han Mao Kiah and Gregory Puleo.)

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**Patrick Solé (CNRS, France)**

**Title:**Good algebraic codes exist

**Abstract:**We survey the algebraic structure and asymptotic performance of the self-dual and LCD classes of quasi-cyclic, quasi-twisted, and dihedral codes over finite fields and finite rings. Of special interest is the case of low index: double circulant codes and four circulant codes. An application to additive cyclic codes is given.

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**Vinay Vaishampayan (City University of New York, USA)**

**Title:**Function Computation in Networked Environments

**Abstract:**We address communication issues that underlie computation of a function f(x,y,z,...) when the variables x, y, z... are known at distinct nodes in a network. This falls under the broad subject of communication complexity, which continues to be the subject of re-search in both the computer science and information theory communities. This problem has broader significance, in areas that are not traditionally studied by information and communication theorists, e.g. theoretical physics and economics. The first part of the talk will address origins of the problem and theoretical developments, in both the computer science and the information theory literature. The second part of the talk will address constructions the problem of building codes for this problem, tracing origins in quantization theory to recent developments. The last part will address some of the current applications that are driving interest in the problem along with open issues. Specifically, we will consider collaborative systems that arise in distributed machine learning and and distributed security. The underlying communication required for collaboration and cooperation, which can often place an unreasonable burden on a wide area network, can be significantly reduced through careful code design. (Based on joint work with M. F. Bollauf and S. I. R. Costa.)

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__ORAL PRESENTATIONS__**Two-Point Codes on a maximal curve X which cannot be covered by the Hermitian curve**

Alonso Castellanos and Arnoldo Herrera

Abstract: We construct a new example the maximal curve $\mathcal{X}$ which cannot be covered by the Hermitian curve. We calculate the Weierstrass Semigroup in the only point at infinity $P_\infty$. We determine the Weierstrass semigroup of a pair of certain rational points on $\mathcal{X}$ and we use theses semigroups to construct one and two-point AG codes.

**A construction of F_2 -linear cyclic, MDS codes**

Sara D. Cardell, Joan-Josep Climent, Daniel Panario and Brett Stevens

Abstract: In this paper we construct of $\Fset_2$-linear codes over $\Fset_{2}^{b}$ with length $n$ and dimension $n-r$ where $n = rb$. These codes have good properties, namely cyclicity, low density parity-check matrices and maximum distance separation. For the construction, we consider an odd prime $p$, let $n=p-1$ and utilize a partition of $\Zset_n$. Then we apply a Zech logarithm to the elements of these sets and use the results to construct an {\em index array} which represents the parity-check matrix of the code. These codes are always cyclic. The density of the parity-check matrix is essentially $1/b$ and thus for a fixed $r$ the density decreases to 0 as $n$ grows. When $r=2$ we can prove that these codes are always maximum distance separable. For higher $r$ some of them retain this property.

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**An example of a non-Abelian group code achieving channel capacity**

Jorge Arpasi

Abstract: In this work is shown analyticaly that the encoding capacity of the 8PSK-AWGN channel achieves the Shannon's channel capacity when the code is over the diedral group D4.

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**The weight distribution of families of reducible cyclic codes through the weight distribution of some irreducible cyclic codes**

Jesús Elisandro Cuén Ramos and Gerardo Vega

Abstract: A cyclic code ${\cal C}$ is reducible if its parity-check polynomial $h(x)$ is at least the product of two irreducible polynomials. Moreover, since cyclic codes are linear, the cyclic code ${\cal C}$ is the direct sum (as vector spaces) of the irreducible cyclic codes whose parity-check polynomials are the irreducible factors of $h(x)$. For this reason, the problem of determining the weight distribution of a reducible cyclic code is, in general, more difficult than determining the weight distribution of an irreducible cyclic code. The purpose of this paper is to reduce the weight distribution calculation for certain families of reducible cyclic codes to the weight distribution calculation of certain irreducible cyclic codes. We achieved this reduction through an old and apparently unnoticed identity among the weight distribution of some cyclic codes. We show that, through it, we can reduce significantly the calculation of the weight distribution of some families of reducible cyclic codes that were recently reported in several works. Moreover, by means of this identity we present the weight distribution of other cyclic codes that had not been reported up to now.

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**Self-orthogonal Codes over $\mathbb{F}_{q}+u\mathbb{F}_{q}$**

Rowena Alma Betty, Fidel Nemenzo and Lucky Erap Galvez

Abstract: In this paper, we establish a mass formula for Euclidean and Hermitian self-orthogonal codes over the finite ring $\mathbb{F}_{q}+u\mathbb{F}_{q}$, where $\mathbb{F}_{q}$ is the finite field of order $q$ and $u^2=0$. We also give a classification of Euclidean and Hermitian self-orthogonal codes over $\mathbb{F}_2 +u\mathbb{F}_2$ and $\mathbb{F}_3 +u\mathbb{F}_3$, up to length 6.

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**On q-analog Steiner systems of rank metric codes**

Javier de La Cruz

Abstract: In this work we prove that rank metric codes with special properties imply the existence of q-analogs of suitable designs. More precisely, we show that the minimum weight vectors of a $[2d, d, d]$ dually almost MRD code $\mathcal{C}\leq \mathbb{F}^{2d}_{q^m}$ $(2d ≤ m)$ which has no code words of rank weight $d + 1$ form a $q$-analog Steiner system $\mathcal{S_q}(d − 1, d, 2d)$. This is the $q$-analog of a result in classical coding theory and may be seen as a first step attempting to prove a q-analog the famous Assmus-Mattson Theorem.

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**Challenge Codes for Physically Unclonable Functions with Gaussian Delays: A Maximum Entropy Problem**

Alexander Schaub, Olivier Rioul, Joseph Boutros, Sylvain Guilley and Jean-Luc Danger

Abstract: Suppose we are given a (nonlinear) (n,M) code C with M codewords c_i, c_i = (+1,-1)^n, and n i.i.d. standard Gaussian variables X = (X_1,X_2,...,X_n). Consider the scalar products between c_i and X, and B_i the sign of this scalar product. The question addressed in this paper is the following. What is the joint entropy of the sign bits H = H(B_1,B_2,...,B_M) ? In particular, can we evaluate the maximum entropy attained for the full universe code ? Despite appearances, this problem turns out to be purely combinatorial as shown below. The motivation for this problem comes from hardware security. Modern secure integrated circuits make use of hardware primitives called physically unclonable functions (PUFs) that can generate unique identifiers from challenges. Having a unique identifier for each physical chip allows one to authenticate it in a secure way. PUFs exploit small, uncontrollable physical variations of the manufacturing process that cannot be replicated, hence the name physically unclonable. Such small variations are often modeled by Gaussian random variables.

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**Construction of Geometrically Uniform Hyperbolic Signal Space Codes**

Cátia Regina De Oliveira Quilles Queiroz and Reginaldo Palazzo

Abstract: In this paper construction of geometrically uniform (GU) hyperbolic signal space codes over the quaternion orders associated with the arithmetic Fuchsian groups Gamma_{4g}, where g is the genus of the associated surface, is proposed. These Fuchsian groups consist of the edge-pairing isometries of the regular hyperbolic polygons (fundamental region) P_{4g}, which tessellate the hyperbolic plane D^{2}. The corresponding tessellations are self-dual tessellations {4g, 4g}. Knowing the generators of the quaternion orders which realize the edge-pairings of the polygons, the signal points of the signal sets derived from the graphs associated with the quotient rings of the quaternion orders are determined.

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**The MDS conjecture, completeness of rational normal curves, and the polynomial method.**

Krishna Kaipa

Abstract: The MDS conjecture asserts that there is no $GF(q)$-linear MDS code of length exceeding $ q+1$ and dimension less than $q$, with some exceptions when q is even. There is a special and simpler case of this conjecture: is there an MDS code of length $q+2$ extending a GRS code (generalized Reed-Solomon) of length $q+1$? A more general problem is to find conditions on $(n,k,q) $such that any MDS extension of a length n and k-dimensional GRS code must itself be GRS. The best general answer to this problem was given by R. Roth and G. Seroussi in 1986. We will present a new proof of this result using the combinatorial nullstellensatz, and discuss the application of this technique to the aforementioned problems.

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**Fuchsian group generators associated with the C_{2,8} channel quantization**

Anderson Oliveira and Reginaldo Palazzo Jr.

Abstract: In this paper we consider the steps to be followed by a designer of a communication system regarding the channel output quantization problem under the topological space approach. The first step is to determine the genus of each surface which this channel may be embedded and the second step is to determine the algebraic structure (Fuchsian group generators) associated with the fundamental region of each surface. The embedding provides an interval of possible values taking on by the genus, denoted by $g$, the topological invariant of a surface. The algebraic structure comes from the fact that for each value of $g$ corresponds a hyperelliptic Riemann surface. For each such a hyperelliptic Riemann surface there is an associated linear second order Fuchsian differential equation whose linearly independent solutions provide the generators of the Fuchsian group, the algebraic structure being sought.

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**Invariant Theory and Self-Dual Codes in the NRT-Metric**

Welington Santos and Marcelo Muniz

Abstract: In this paper we will use the polynomial invariant theory to describe the $H$-enumerator of a self-dual code in the Niederreiter-Rosenbloom-Tsfasman metric. The results of this paper also describe the shape enumerator of a linear self-dual ordered code defined by Barg.

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**Hyperbolic Color Codes on Densest Tessellations**

Clarice Albuquerque and Reginaldo Palazzo Jr.

Abstract: In this work we propose a construction of color code on two-dimensional manyfolds with genus $g \ge 2$ considering the denser regular tessellations of the hyperbolic plane. Considering the $\{12i-6,3\}$ tessellations, for $i=2,3,\ldots$, we obtain a class $[[\frac{(6+12i)(g-1)}{3(i-1)}, 4g, d]]$ whose encoding rate is assintotically 1.

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**Metrics that respect the support**

Roberto Machado

Abstract: In this work we explore the family of metrics determined by $S$-weights, i.e., non-negative functions over finite fields that respect the support. First, we introduce a conditional sum of weights and classify those which every set of equivalent weights is closed under such sums. Then, we introduce an structure to represent all decision criteria which allows us to characterize the group of linear isometries for $S$-weights sharing the same equivalence class regarding the decoding criterion.

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**The Covering Problem in Hierarchical Poset Spaces over Finite Rings**

Marcos V. P. Spreafico and Otavio Dos Santos

Abstract: We investigate the minimum size of a Covering code and a Short Covering code in a hierarchical poset space. We shown that both problems are closely related with its correspondents in Hamming space.

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**Lattices for Cryptography**

Jheyne Nayara Ortiz, Robson Ricardo Araujo, Ricardo Dahab, Diego Aranha and Sueli Costa

Abstract: In this survey paper, we discuss properties of lattices from a cryptographic perspective. We clarify relations between Geometry, Algebraic Number Theory and Cryptography, with focus on the application of lattices to encoding and encrypting sensitive data. In Lattice-based Cryptography, these relations are described by cryptographic schemes whose security relies on hard instances of computational problems over lattices. We address here lattice-based and related problems and the choice of system parameters to securely instantiate them.

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**Construction of Quantum Subspace Codes of the Simplex Type for Quantum Network Coding**

Wanessa Gazzoni, Gabriella Miyamoto and Reginaldo Palazzo Jr.

Abstract: In this paper, we propose a construction of geometrically uniform subspace codes. First, we show the construction. After that, we label these codes with classical codes and finally we show the relation between the constructed codes and quantum codes, obtaining that the constructed codes are quantum subspaces codes of simplex type.

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**Quantum-resistant Cryptography based on Isogenies between Elliptic Curves**

João Paulo Da Silva, Ricardo Dahab and Julio López

Abstract: The security of public-key systems is based on the difficulty of solving certain mathematical problems. With the possible emergence of large scale quantum computers several of theses problems, such as factoring and computing discrete logarithms, would be efficiently solved. Research on quantum-resistant public-key cryptography, also called post-quantum cryptography (PQC), has been productive in recent years. Several newly developed algorithms appear to be good candidates for the next generation of public-key cryptography standards. In particular, public-key cryptosystems based on the problem of computing isogenies between supersingular elliptic curves were recently proposed as strong candidate in PQC. In this work we have surveyed recent applications of isogenies to construct very interesting cryptographic primitives for PQC.

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**Non-existence of linear e-perfect Lee codes depending on the base-3-representation of e**

Claudio Qureshi

Abstract: The Golomb-Welch conjecture states that there are no $e$-perfect Lee codes in $\Z^n$ for $n\geq 3$ and $e\geq 2$. This conjecture remains open even for linear codes. Some recent results establish the non-existence of linear $e$-perfect Lee codes in $\Z^n$ for infinitely many dimensions $n$, for $e=2,3$ and $4$. In this paper we prove that this is true for almost all $e$ (i.e. a subset of positive integers with density $1$). For instance, if $e$ contains a digit $1$ in its base-$3$ representation which is not in the unit place, there are no linear $e$-perfect Lee codes in $\Z^n$ for infinitely many dimensions $n$.

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**A shape-gain approach for vector quantization based on flat tori and dual lattices A*_n**

Fabiano Boaventura de Miranda and Cristiano Torezzan

Abstract: In this paper we present a vector quantization framework for Gaussian sources combining a spherical code in layers of flat tori and the shape-gain technique. The basic concepts of spherical codes in tori layers are reviewed and an specific construction is proposed for the quantization of the shape using the $k/2$-dimensional dual lattice $A^{*}_{k/2}$ as its pre-image. A scalar quantizer is optimized for the gain using the Lloyd-Max algorithm for a given rate. The computational complexity is dominated by the lattice decoding process, which is carried out in the $k/2$ dimension. In the case of using a $A^{*}_{k/2}$ lattice, the lattice decoding can be done in $O(n^2\log{n})$. The proposed quantizer is described in details and some numerical results are presented for dimensions $k=\{4,6,8\}$.

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**Perfect Codes and Generalized Ordinal Sum of Posets**

Luciano Panek, Jerry Anderson Pinheiro, Marcelo Firer and Marcelo Muniz Alves

Abstract: We consider on $\mathbb{F}_{q}^{n}$ the metric determined by a poset $P$ over [n]. Given a perfect code $C\subseteq \mathbb{F}_{q}^{n}$, we show a way to extend it into a perfect code on $\mathbb{F}_{q}^{n+m}$, for some posets over $[n + m]$ determined by a poset $Q$ over $[m]$ and some relations between $[n]$ and $[m]$.

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**Information theory associated to Tsallis' 2-entropy**

Juan Pablo Vigneaux

Abstract: While Shannon entropy is related to the exponential growth of multinomial coefficients, we show that Tsallis 2-entropy is connected to their $q$-version; when $q$ is a prime power, they count the number of flags in $\mathbb{F}_q^n$ with prescribed length and dimensions ($\mathbb{F}_q$ denotes the field of order $q$). In particular, the $q$-binomial coefficient ${n \brack k}$ counts vector subspaces of dimension $k$. Based on this idea, we obtain a combinatorial interpretation for non-additivity. We consider statistical systems whose configurations are described by flags: they provide a frequentist justification for the maximum entropy principle with Tsallis statistics. We introduce then a discrete-time stochastic process associated to the $q$-binomial distribution, that generates at time $n$ a vector subspace of $\mathbb{F}_q^n$. The concentration of measure on certain “typical subspace” allows us to extend to this setting the asymptotic equipartition property as well as its applications to compression and coding.

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**Algebraic QC-LDPC Codes with Girth 6 and Free of Small Size Elementary Trapping Sets**

Farzane Amirzade, Mohammad Reza Sadeghi and Daniel Panario

Abstract: Trapping sets significantly influence the performance of low-density parity-check codes. An elementary trapping set (ETS) causes high decoding failure rate and exert a strong influence on the error floor of the code. In this paper, our goal is to construct fully connected $(3,n)$-regular QC-LDPC codes with girth 6 whose Tanner graphs are free of ETSs of small size. To reach our goal, first, we present an algebraic structure for the exponent matrix of QC-LDPC codes with girth at least 6. Then, we obtain sufficient conditions for an exponent matrix to have a Tanner graph which is free of some ETSs with small size.

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**A generalization of Quantum Relative Entropy**

Luiza Helena Félix de Andrade, Rui Facundo Vigelis and Charles Casimiro Cavalcante

Abstract: We propose a generalization of quantum relative entropy by the use of a deformed exponential $\varphi$ and which allows some flexibility in the statistical model we show some properties of this proposed relative entropy.

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**Completely Galois-disjoint polycyclic codes over finite chain rings**

Thomas Blackford, Alexandre Fotue Tabue and Edgar Martínez-Moro

Abstract: Galois images of polycyclic codes over a finite chain ring $S$ of invariants $(q^m,s)$ whose period is relatively prime to $q$ and their dual codes are investigated. The characterization of completely Galois-disjoint polycyclic codes, is established. The restriction code and trace code of any polycyclic code, are also determined.

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