arXiv subjects

Edgar Martínez-Moro

Publications and source records attributed to Edgar Martínez-Moro.

35 records · Page 2Linked to original sources

arXiv2022

The hull of two classical propagation rules and their applications

In this work, we study and determine the dimensions of Euclidean and Hermitian hulls of two classical propagation rules, namely, the direct sum construction and the $(\mathbf{u},\mathbf{u+v})$-construction. Some new criteria for the resulting codes derived from these two propagation rules being self-dual, self-orthogonal, or linear complementary dual (LCD) codes are given. As an application, we construct some linear codes with prescribed hull dimensions, many new binary, ternary Euclidean formally self-dual (FSD) LCD codes, and quaternary Hermitian FSD LCD codes. Some new even-like, odd-like, Euclidean and Hermitian self-orthogonal codes are also obtained. Many of {these} codes are also (almost) optimal according to the Database maintained by Markus Grassl. Our methods contribute positively to improve the lower bounds on the minimum distance of known LCD codes.

cs.IT↗

arXiv2022

Private Information Retrieval Schemes Using Cyclic Codes

A Private Information Retrieval (PIR) scheme allows users to retrieve data from a database without disclosing to the server information about the identity of the data retrieved. A coded storage in a distributed storage system with colluding servers is considered in this work, namely the approach in [$t$-private information retrieval schemes using transitive codes, IEEE Trans. Inform. Theory, vol. 65, no. 4, pp. 2107-2118, 2019] which considers a storage and retrieval code with a transitive group and provides binary PIR schemes with the highest possible rate. Reed-Muller codes were considered in [$t$-private information retrieval schemes using transitive codes, IEEE Trans. Inform. Theory, vol. 65, no. 4, pp. 2107-2118, 2019]. In this work, we consider cyclic codes and we show that binary PIR schemes using cyclic codes provide a larger constellation of PIR parameters and they may outperform the ones coming from Reed-Muller codes in some cases.

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arXiv2022

Public key cryptography based on skew dihedral group rings

In this paper, we propose to use a skew dihedral group ring given by the group $D_{2n}$ and the finite field $\mathbb{F}_{q^2}$ for public-key cryptography. Using the ambient space $\mathbb{F}_{q^{2}}^θ D_{2n}$ and a group homomorphism $θ: D_{2n} \rightarrow \mathrm{Aut}(\mathbb{F}_{q^2})$, we introduce a key exchange protocol and present an analysis of its security. Moreover, we explore the properties of the resulting skew group ring $\mathbb{F}_{q^{2}}^θ D_{2n}$, exploiting them to enhance our key exchange protocol. We also introduce a probabilistic public-key scheme derived from our key exchange protocol and obtain a key encapsulation mechanism (KEM) by applying a well-known generic transformation to our public-key scheme. Finally, we present a proof-of-concept implementation of our cryptographic constructions. To the best of our knowledge, this is the first paper that proposes a skew dihedral group ring for public-key cryptography.

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arXiv2022

Free Resolutions and Generalized Hamming Weights of binary linear codes

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.

cs.IT↗

arXiv2022

Galois LCD codes over mixed alphabets

We study (Galois) linear complementary dual codes over mixed alphabets arising from finite chain rings. We give a characterization of when a given code is of We study (Galois) linear complementary dual codes over mixed alphabets arising from finite chain rings. We give a characterization of when a given code is of this type and when it is Galois invariant. Finally, this leads to a study of the Gray image of $\mathbb{F}_p\mathbb{F}_p[θ]$-linear codes, where $p\in\{2; 3\}$ and $θ\neqθ^2=0$, that provides $\mathbb{F}_p$-linear complementary dual codes.

cs.IT↗

arXiv2021

Cyclic codes over a non-chain ring $R_{e,q}$ and their application to LCD codes

Let $\mathbb{F}_q$ be a finite field of order $q$, a prime power integer such that $q=et+1$ where $t\geq 1,e\geq 2$ are integers. In this paper, we study cyclic codes of length $n$ over a non-chain ring $R_{e,q}=\mathbb{F}_q[u]/\langle u^e-1\rangle$. We define a Gray map $φ$ and obtain many { maximum-distance-separable} (MDS) and optimal $\mathbb{F}_q$-linear codes from the Gray images of cyclic codes. Under certain conditions we determine { linear complementary dual} (LCD) codes of length $n$ when $\gcd(n,q)\neq 1$ and $\gcd(n,q)= 1$, respectively. It is proved that { a} cyclic code $\mathcal{C}$ of length $n$ is an LCD code if and only if its Gray image $φ(\mathcal{C})$ is an LCD code of length $4n$ over $\mathbb{F}_q$. 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 $R_{e,q}$.

cs.IT↗

arXiv2021

A transform approach to polycyclic and serial codes over rings

In this paper, a transform approach is used for polycyclic and serial codes over finite local rings in the case that the defining polynomials have no multiple roots. This allows us to study them in terms of linear algebra and invariant subspaces as well as understand the duality in terms of the transform domain. We also make a characterization of when two polycyclic ambient spaces are Hamming-isometric.

cs.IT↗

arXiv2021

Galois hulls of cyclic serial codes over a finite chain ring

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 $\mathbb{F}_{p^r}$ is the residue field of $R$, and we provide some results of its relative growth.

cs.IT↗

arXiv2020

Linear Complementary Pair Of Group Codes over Finite Chain Rings

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 $d(D^\bot)$. It has been recently shown that if $C$ and $D$ are both 2-sided group codes over a finite field, then $C$ and $D^\bot$ 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.

cs.IT↗

arXiv2020

Lifted Codes and Lattices from Codes Over Finite Chain Rings

In this paper we give the generalization of lifted codes over any finite chain ring. This has been done by using the construction of finite chain rings from $p$-adic fields. Further we propose a lattice construction from linear codes over finite chain rings using lifted codes.

cs.IT↗

arXiv2019

Do non-free LCD codes over finite commutative Frobenius rings exist?

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 $\mathbb{Z}_4$ codes of different lengths {which are} cyclic LCD codes over $\mathbb{Z}_4$.

cs.IT↗

arXiv2019

Vardøhus Codes: Polar Codes Based on Castle Curves Kernels

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.

cs.IT↗

arXiv2018

On polycyclic codes over a finite chain ring

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.

cs.IT↗

arXiv2017

Additive cyclic codes over finite commutative chain rings

Additive cyclic codes over Galois rings were investigated in previous works. 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 previous studies, 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.

cs.IT↗

arXiv2016

On Counting Subring-Subcodes of Free Linear Codes Over Finite Principal Ideal Rings

Let $R$ be a finite principal ideal ring and $S$ the Galois extension of $R$ of degree $m$. For $k$ and $k_0$, positive integers we determine the number of free $S$-linear codes $B$ of length $l$ with the property $k = rank_S(B)$ and $k_0 = rank_R (B\cap R^l)$. This corrects a wrong result which was given in the case of finite fields.

cs.IT↗

arXiv2015

On the ideal associated to a linear code

This article aims to explore the bridge between the algebraic structure of a linear code and the complete decoding process. To this end, we associate a specific binomial ideal $I_+(\mathcal C)$ to an arbitrary linear code. The binomials involved in the reduced Gröbner basis of such an ideal relative to a degree-compatible ordering induce a uniquely defined test-set for the code, and this allows the description of a Hamming metric decoding procedure. Moreover, the binomials involved in the Graver basis of $I_+(\mathcal C)$ provide a universal test-set which turns out to be a set containing the set of codewords of minimal support of the code.

cs.IT↗

arXiv2012

On the Composition of Secret Sharing Schemes Related to Codes

In this paper we construct a subclass of the composite access structure introduced by Martínez et al. based on schemes realizing the structure given by the set of codewords of minimal support of linear codes. This class enlarges the iterated threshold class studied in the same paper. Furthermore all the schemes on this paper are ideal (in fact they allow a vector space construction) and we arrived to give a partial answer to a conjecture stated in the above paper. Finally, as a corollary we proof that all the monotone access structures based on all the minimal supports of a code can be realized by a vector space construction.

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