ETD

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Tesi etd-09052017-191245


Tipo di tesi
Tesi di laurea magistrale
Autore
DE SALVO, STEFANO
URN
etd-09052017-191245
Titolo
Twisted Gabidulin codes and new Maximum Rank Distance codes
Dipartimento
MATEMATICA
Corso di studi
MATEMATICA
Relatori
relatore Prof. Del Corso, Ilaria
relatore Prof. Rosenthal, Joachim
correlatore Neri, Alessandro
Parole chiave
  • codes
  • rank metric
  • Gabidulin
  • Twisted
  • Maximum Rank Distance
  • MRD
Data inizio appello
22/09/2017
Consultabilità
Completa
Riassunto
Rank-metric codes have been introduced by Delsarte and then rediscovered by Gabidulin. Since then, rank-metric codes have been used in many applications, in particular for network coding. For these codes, there exists a Singleton-type bound and codes that attain this bound are called Maximum Rank Distance (MRD) codes. The first family of MRD codes was constructed by Delsarte and, indipendently, by Gabidulin and was later generalized by Kshevetskiy and Gabidulin. That is the family of Gabidulin codes. In our work, we focus on F_{q^m}-subspaces of F_{q^m}^n that we call linear rank-metric codes. A linear rank-metric code that is also MRD is a linear MRD code. It has been proved hat almost every linear rank-metric code is MRD but not many general constructions of MRD codes are known. Sheekey constructed a new family of MRD codes that was later generalized by Lunardon, Trombetti and Zhou. These MRD codes are called generalized twisted Gabidulin codes. Some generalized twisted Gabidulin codes are linear codes. The main aim of our work is to characterize all linear MRD codes over F_{q^m} that are contained in the generalized Gabidulin code of dimension k+1, parameter s and support g=(g_1,g_2,...,g_n) and that contain the generalized Gabidulin code of dimension k-1, parameter s and support g^(q^s)=(g_1^(q^s),g_2^(q^s),...,g_n^(q^s)). We managed to find sets of parameters k, n, m for which the class of codes found by Sheekey et al. include all the rank-metric codes that satisfy the properties above. Furthermore, we also found different sets of parameters for which they are not the only codes satisfying such properties.
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