arXiv subjects

Marinês Guerreiro

Publications and source records attributed to Marinês Guerreiro.

5 recordsLinked to original sources

arXiv2018

Abelian Noncyclic Orbit Codes and Multishot Subspace Codes

In this paper we characterize the orbit codes as geometrically uniform codes. This characterization is based on the description of all isometries over a projective geometry. In addition, the Abelian orbit codes are defined and a new construction of Abelian non-cyclic orbit codes is presented. In order to analyze their structures, the concept of geometrically uniform partitions have to be reinterpreted. As a consequence, a substantial reduction in the number of computations needed to obtain the minimum subspace distance of these codes is achieved and established. An application of orbit codes to multishot subspace codes obtained according to a multi-level construction is provided.

cs.IT↗

arXiv2015

Group Algebra and Coding Theory

Group algebras have been used in the context of Coding Theory since the beginning of the latter, but not in its full power. The work of Ferraz and Polcino Milies entitled Idempotents in group algebras and minimal abelian codes (Finite Fields and their Applications, 13, (2007) 382-393) gave origin to many thesis and papers linking these two subjects. In these works, the techniques of group algebras are mainly brought into play for the computing of the idempotents that generate the minimal codes and the minimum weight of such codes. In this paper I summarize the main results of the work done by doctorate students and research partners of Polcino Milies and Ferraz.

math.RA↗

arXiv2012

Exceptional representations of simple algebraic groups in prime characteristic

Let G be a simply connected simple algebraic group over an algebraically closed field K of characteristic p>0 with root system R, and let ${\mathfrak g}={\cal L}(G)$ be its restricted Lie algebra. Let V be a finite dimensional ${\mathfrak g}$-module over K. For any point $v\inV$, the {\it isotropy subalgebra} of $v$ in $\mathfrak g$ is ${\mathfrak g}_v=\{x\in{\mathfrak g}/x\cdot v=0\}$. A restricted ${\mathfrak g}$-module V is called exceptional if for each $v\in V$ the isotropy subalgebra ${\mathfrak g}_v$ contains a non-central element (that is, ${\mathfrak g}_v\not\subseteq {\mathfrak z(\mathfrak g)}$). This work is devoted to classifying irreducible exceptional $\mathfrak g$-modules. A necessary condition for a $\mathfrak g$-module to be exceptional is found and a complete classification of modules over groups of exceptional type is obtained. For modules over groups of classical type, the general problem is reduced to a short list of unclassified modules. The classification of exceptional modules is expected to have applications in modular invariant theory and in classifying modular simple Lie superalgebras.

math.RT↗

arXiv2012

Minimal Binary Abelian Codes of length $p^m q^n$

We consider binary abelian codes of length $p^m q^n$, where $p$ and $q$ are prime rational integers under some restrictive hypotheses. In this case, we determine the idempotents generating minimal codes and either the respective weights or bounds of these weights. We give examples showing that these bounds are attained in some cases.

cs.IT↗

arXiv2012

G-equivalence in group algebras and minimal abelian codes

Let G be a finite abelian group and F a field such that char(F) does not divide |G|. Denote by FG the group algebra of G over F. A (semisimple) abelian code is an ideal of FG. Two codes I and J of FG are G-equivalent if there exists an automorphism of G whose linear extension to FG maps I onto J In this paper we give a necessary and sufficient condition for minimal abelian codes to be G-equivalent and show how to correct some results in the literature.

cs.IT↗