Edgar Martinez-Moro - Academia.edu (original) (raw)

Papers by Edgar Martinez-Moro

arXiv (Cornell University), Jan 30, 2019

In this paper, we clarify some aspects on LCD codes in the literature. We first prove that a non-... more In this paper, we clarify some aspects on LCD codes in the literature. We first prove that a non-free LCD code does not exist over finite commutative Frobenius local rings. We then obtain a necessary and sufficient condition for the existence of LCD code over finite commutative Frobenius rings. We later show that a free constacyclic code over finite chain ring is LCD if and only if it is reversible, and also provide a necessary and sufficient condition for a constacyclic code to be reversible over finite chain rings. We illustrate the minimum Lee-distance of LCD codes over some finite commutative chain rings and demonstrate the results with examples. We also got some new optimal 4 codes of different lengths which are cyclic LCD codes over 4 .

$Research paper thumbnail of Vard{\o}hus Codes: Polar Codes Based on Castle Curves Kernels$

arXiv (Cornell University), Jan 21, 2019

In this paper we show some applications of algebraic curves to the construction of kernels of pol... more In this paper we show some applications of algebraic curves to the construction of kernels of polar codes over a discrete memoryless channel which is symmetric w.r.t the field operations. We will also study the minimum distance of the polar codes proposed, their duals and the exponents of the matrices used for defining them. All the restrictions that we make to our curves will be accomplished by the so called Castle Curves.

arXiv (Cornell University), Dec 7, 2016

Let R be a finite principal ideal ring and S the Galois extension of R of degree m. For k and k ′... more Let R be a finite principal ideal ring and S the Galois extension of R of degree m. For k and k ′ , positive integers we determine the number of free S-linear codes B of length ℓ with the property k = rank S (B) and k ′ = rank R (B ∩ R ℓ). This corrects a wrong result [1, Theorem 6] which was given in the case of finite fields.

arXiv (Cornell University), Jul 11, 2019

A locally recoverable code is an error-correcting code such that any erasure in a single coordina... more A locally recoverable code is an error-correcting code such that any erasure in a single coordinate of a codeword can be recovered from a small subset of other coordinates. In this article we develop an algorithm that computes a recovery structure as concise posible for an arbitrary linear code C and a recovery method that realizes it. This algorithm also provides the locality and the dual distance of C. Complexity issues are studied as well. Several examples are included.

Contemporary mathematics, 2015

arXiv (Cornell University), Mar 31, 2022

In this work, we explore the relationship between free resolution of some monomial ideals and Gen... more In this work, we explore the relationship between free resolution of some monomial ideals and Generalized Hamming Weights (GHWs) of binary codes. More precisely, we look for a structure smaller than the set of codewords of minimal support that provides us some information about the GHWs. We prove that the first and second generalized Hamming weight of a binary linear code can be computed (by means of a graded free resolution) from a set of monomials associated to a binomial ideal related with the code. Moreover, the remaining weights are bounded by the Betti numbers for that set.

Finite Fields and Their Applications, 2022

In this paper we explore some properties of Galois hulls of cyclic serial codes over a chain ring... more In this paper we explore some properties of Galois hulls of cyclic serial codes over a chain ring and we devise an algorithm for computing all the possible parameters of the Euclidean hulls of that codes. We also establish the average p r-dimension of the Euclidean hull, where F p r is the residue field of R, and we provide some results of its relative growth.

Designs, Codes and Cryptography, Jul 4, 2020

We are proud to present this special issue in honor of Ruud Pellikaan. It presents 11 selected pa... more We are proud to present this special issue in honor of Ruud Pellikaan. It presents 11 selected papers out of 27 submitted by the participants as postconference contributions of the workshop entitled "Codes, Cryptology and Curves in honour of Ruud Pellikaan" that took place at Eindhoven University of Technology (TU/e) 7-9 March of 2019. The workshop aimed to bring together researchers at all levels and career stages, providing an opportunity to share and discuss recent progress to celebrate Ruud's vast contribution to the fields of Coding Theory, Cryptology and Curves (Algebraic Geometry). There was a warm and supportive atmosphere during the meeting which showed the high esteem in which Ruud is held in the international mathematical community. As it is impossible to adequately cover all of the recent advances based on Ruud's works in a single issue, we intend to present a fragment of the timely research topics and noteworthy new developments related to his research. As we mentioned before, we received many more papers than we could include and we thank the authors of all the papers submitted. Due to the high quality of submissions, the process of deciding which papers to include was not easy and we are very grateful to the referees for their assistance in helping us reach our final selection. We also highly appreciate the journal and Springer Nature support during the entire process, specially the helpful support of Shamima Banu Rajesh who was available to us all the time. We, Ruud's friends, colleagues, co-authors and students, have been privileged to benefit from his inspiration, hard work, dedication and generosity to his friends, and his contributions B Edgar Martínez-Moro

Designs, Codes and Cryptography, Dec 8, 2017

Let R be a finite commutative Frobenius ring and S a Galois extension of R of degree m. For posit... more Let R be a finite commutative Frobenius ring and S a Galois extension of R of degree m. For positive integers k and k , we determine the number of free S-submodules B of S with the property k = rank S (B) and k = rank R (B ∩ R). This corrects the wrong result (Bill in Linear Algebr Appl 22:223-233, 1978, Theorem 6) which was given in the language of codes over finite fields.

ACM Communications in Computer Algebra, Jun 1, 2022

$Research paper thumbnail of Cyclic codes over a non-chain ring <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>R</mi><mrow><mi>e</mi><mo separator="true">,</mo><mi>q</mi></mrow></msub></mrow><annotation encoding="application/x-tex">R_{e,q}</annotation></semantics></math>Re,q and their application to LCD codes$

arXiv (Cornell University), Jun 15, 2021

Let F q be a finite field of order q, a prime power integer such that q = et + 1 where t ≥ 1, e ≥... more Let F q be a finite field of order q, a prime power integer such that q = et + 1 where t ≥ 1, e ≥ 2 are integers. In this paper, we study cyclic codes of length n over a non-chain ring R e,q = F q [u]/ u e −1. We define a Gray map ϕ and obtain many maximum-distance-separable (MDS) and optimal F q-linear codes from the Gray images of cyclic codes. Under certain conditions we determine linear complementary dual (LCD) codes of length n when gcd(n, q) = 1 and gcd(n, q) = 1, respectively. It is proved that a cyclic code C of length n is an LCD code if and only if its Gray image ϕ(C) is an LCD code of length 4n over F q. Among others, we present the conditions for existence of free and non-free LCD codes. Moreover, we obtain many optimal LCD codes as the Gray images of non-free LCD codes over R e,q .

arXiv (Cornell University), Feb 13, 2021

arXiv (Cornell University), Dec 6, 2019

Linear complementary dual (LCD) codes and linear complementary pair (LCP) of codes over finite fi... more Linear complementary dual (LCD) codes and linear complementary pair (LCP) of codes over finite fields have been intensively studied recently due to their applications in cryptography, in the context of side channel and fault injection attacks. The security parameter for an LCP of codes (C, D) is defined as the minimum of the minimum distances d(C) and d(D ⊥). It has been recently shown that if C and D are both 2-sided group codes over a finite field, then C and D ⊥ are permutation equivalent. Hence the security parameter for an LCP of 2-sided group codes (C, D) is simply d(C). We extend this result to 2-sided group codes over finite chain rings.

arXiv (Cornell University), May 24, 2005

The structure of multivariate semisimple codes over a finite chain ring RRR is established using ... more The structure of multivariate semisimple codes over a finite chain ring RRR is established using the structure of the residue field barR\bar RbarR. Multivariate codes extend in a natural way the univariate cyclic and negacyclic codes and include some non-trivial codes over RRR. The structure of the dual codes in the semisimple abelian case is also derived and some conditions on the existence of selfdual codes over RRR are studied.

$Research paper thumbnail of Additive semisimple multivariable codes over <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow></mrow><annotation encoding="application/x-tex"></annotation></semantics></math>{\mathbb{F}_4}$$$

Designs, Codes and Cryptography, Mar 22, 2012

The structure of additive multivariable codes over F 4 (the Galois field with 4 elements) is pres... more The structure of additive multivariable codes over F 4 (the Galois field with 4 elements) is presented. The semisimple case (i.e., when the defining polynomials of the code have no repeated roots) is specifically addressed. These codes extend in a natural way the abelian codes, of which additive cyclic codes of odd length are a particular case. Duality of these codes is also studied.

arXiv (Cornell University), Jan 23, 2017

Additive cyclic codes over Galois rings were investigated in [3]. In this paper, we investigate t... more Additive cyclic codes over Galois rings were investigated in [3]. In this paper, we investigate the same problem but over a more general ring family, finite commutative chain rings. When we focus on non-Galois finite commutative chain rings, we observe two different kinds of additivity. One of them is a natural generalization of the study in [3], whereas the other one has some unusual properties especially while constructing dual codes. We interpret the reasons of such properties and illustrate our results giving concrete examples.

HAL (Le Centre pour la Communication Scientifique Directe), Aug 1, 2016

We can associate to each linear code C defined over a finite field the matroid M[H] of its parity... more We can associate to each linear code C defined over a finite field the matroid M[H] of its parity check matrix H. For any matroid M one can define its generalized Hamming weights which are the same as those of the code C. In [2] the authors show that the generalized Hamming weights of a matroid are determined by the N-graded Betti numbers of the Stanley-Reisner ring of the simplicial complex whose faces are the independent set of M. In this talk we go a step further. Our practical results indicate that the generalized Hamming weights of a linear code C can be obtained from the monomial ideal associated with a test-set for C. Moreover, recall that in [3] we use the Gröbner representation of a linear code C to provide a test-set for C. Our results are still a work in progress, but its applications to Coding Theory and Cryptography are of great value.

arXiv (Cornell University), Nov 19, 2018

Galois images of polycyclic codes over a finite chain ring S and their annihilator dual are inves... more Galois images of polycyclic codes over a finite chain ring S and their annihilator dual are investigated. The case when a polycyclic codes is Galois-disjoint over the ring S, is characterized and, the trace codes and restrictions of free polycyclic codes over S are also determined givind an analogue of Delsarte theorem among trace map, any S-linear code and its annihilator dual.

arXiv (Cornell University), Nov 23, 2012

In this paper we use the Gröbner representation of a binary linear code C to give efficient algor... more In this paper we use the Gröbner representation of a binary linear code C to give efficient algorithms for computing the whole set of coset leaders, denoted by CL(C) and the set of leader codewords, denoted by L(C). The first algorithm could be adapted to provide not only the Newton and the covering radius of C but also to determine the coset leader weight distribution. Moreover, providing the set of leader codewords we have a test-set for decoding by a gradient-like decoding algorithm. Another contribution of this article is the relation stablished between zero neighbours and leader codewords.

Notations absences et qui ont toujours trouver un mot d'encouragement. S'il est une chose que je ... more Notations absences et qui ont toujours trouver un mot d'encouragement. S'il est une chose que je peux affirmer, c'est d'avoir noué de belles amitiés partout où je suis allée, j'ai beaucoup de chance. Ed infine, ma non per importanza, ai miei amici, i quali sono sempre stati comprensivi quando non ho potuto essere presente e mi hanno sempre incoraggiata nei momenti piú difficili. Se posso vantarmi di qualcosa, è di aver stretto grandi amicizie durante questo percorso di studio e di questo mi ritengo molto fortunata. Tot slot, maar zeker niet onbelangrijk, voor mijn vrienden, die in staat zijn geweest mij te vergeven voor mijn afwezigheid en altijd een bemoedigend woord hadden. Ik prijs mezelf gelukkig dat ik goede vrienden heb gemaakt tijdens al mijn verblijven. Und nicht zuletzt danke ich meinen Freunden, die mir mein Fehlen immer verziehen und mich stets ermutigt haben. Wenn ich mit Recht behaupten kann, dass ich während aller meiner Aufenthalte wunderbare Freunde gefunden habe, so kann ich mich sehr glücklich schätzen.