DSpace Comunidad:http://hdl.handle.net/10045/339642024-03-28T21:34:59Z2024-03-28T21:34:59ZEquivalence for flag codesNavarro-Pérez, Miguel ÁngelSoler-Escrivà, Xarohttp://hdl.handle.net/10045/1415562024-03-21T01:22:57Z2024-03-07T00:00:00ZTítulo: Equivalence for flag codes
Autor/es: Navarro-Pérez, Miguel Ángel; Soler-Escrivà, Xaro
Resumen: Given a finite field Fq and a positive integer n, a flagis a sequence of nested Fq-subspaces of a vector space Fnq and a flag code is a nonempty collection of flags. The projected codes of a flag code are the constant dimension codes containing all the subspaces of prescribed dimensions that form the flags in the flag code. In this paper we address the notion of equivalence for flag codes and explore in which situations such an equivalence can be reduced to the equivalence of the corresponding projected codes. In addition, this study leads to new results concerning the automorphism group of certain families of flag codes, some of them also introduced in this paper.2024-03-07T00:00:00ZA new invariant for cyclic orbit flag codesAlonso-González, ClementaNavarro-Pérez, Miguel Ángelhttp://hdl.handle.net/10045/1414842024-03-15T01:11:23Z2024-01-22T00:00:00ZTítulo: A new invariant for cyclic orbit flag codes
Autor/es: Alonso-González, Clementa; Navarro-Pérez, Miguel Ángel
Resumen: In the network coding framework, given a prime power q and the vector space Fnq, a constant type flag code is a set of nested sequences of Fq-subspaces (flags) with the same increasing sequence of dimensions (the type of the flag). If a flag code arises as the orbit under the action of a cyclic subgroup of the general linear group over a flag, we say that it is a cyclic orbit flag code. Among the parameters of such a family of codes, we have its best friend, that is the largest field over which all the subspaces in the generating flag are vector spaces. This object permits to compute the cardinality of the code and estimate its minimum distance. However, as it occurs with other absolute parameters of a flag code, the information given by the best friend is not complete in many cases due to the fact that it can be obtained in different ways. In this work, we present a new invariant, the best friend vector, that captures the specific way the best friend can be unfolded. Furthermore, throughout the paper we analyse the strong underlying interaction between this invariant and other parameters such as the cardinality, the flag distance, or the type vector, and how it conditions them. Finally, we investigate the realizability of a prescribed best friend vector in a vector space.2024-01-22T00:00:00ZWeighted Reed–Solomon convolutional codesAlfarano, Gianira N.Napp, DiegoNeri, AlessandroRequena Arévalo, Verónicahttp://hdl.handle.net/10045/1414442024-03-14T01:16:09Z2023-01-24T00:00:00ZTítulo: Weighted Reed–Solomon convolutional codes
Autor/es: Alfarano, Gianira N.; Napp, Diego; Neri, Alessandro; Requena Arévalo, Verónica
Resumen: In this paper we present a concrete algebraic construction of a novel class of convolutional codes. These codes are built upon generalized Vandermonde matrices and therefore can be seen as a natural extension of Reed–Solomon block codes to the context of convolutional codes. For this reason we call them weighted Reed–Solomon (WRS) convolutional codes. We show that under some constraints on the defining parameters these codes are Maximum Distance Profile (MDP), which means that they have the maximal possible growth in their column distance profile. We study the size of the field needed to obtain WRS convolutional codes which are MDP and compare it with the existing general constructions of MDP convolutional codes in the literature, showing that in many cases WRS convolutional codes require significantly smaller fields.2023-01-24T00:00:00ZOn the construction of MRD convolutional codesNapp, DiegoPinto, RaquelSantana, FilipaVela, Carloshttp://hdl.handle.net/10045/1404592024-03-07T01:15:11Z2024-01-22T00:00:00ZTítulo: On the construction of MRD convolutional codes
Autor/es: Napp, Diego; Pinto, Raquel; Santana, Filipa; Vela, Carlos
Resumen: The problem of building optimal block codes, such as MDS codes, over small fields has been an active area of research that led to several interesting conjectures. In the context of convolutional codes, optimal constructions, such as MDS or MDP, are very rare and all require very large finite fields. In this work, we focus on the problem of constructing optimal convolutional codes with respect to the rank distance, i.e. we study the construction of Maximum Rank Distance (MRD) convolutional codes. Considering convolutional codes within a very general framework, we present concrete novel classes of MRD convolutional codes for a large set of given parameters.2024-01-22T00:00:00Z