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Group rings in coding theory
Horáček, Jan ; Žemlička, Jan (advisor) ; Šťovíček, Jan (referee)
This work is focused on linear error-correcting codes in group rings. The basic introduc- tion to group rings and to coding in group rings is given. By code we mean a R-submodul, which generalizes the definition of the code as an ideal. We describe unit-derived and zero- divisor codes. We test paramaters of unit-derived codes. The construction of LDPC codes without short cycles is explained. Except from the derivation of the generator and check matrix we focus on algebraic properties of codes and group rings. We also deal with the self-dual codes, reversible codes or number of units in a finite group algebra of a cyclic group. 1
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Max rings
Beneš, Daniel ; Žemlička, Jan (advisor) ; Šaroch, Jan (referee)
Topic of this thesis is max rings, which are the rings, whose nonzero modu- les have maximal submodules. At the begining we prove a characterization of commutative max rings as rings with T-nilpotent Jacobson radical and von Ne- umann regular factor ring of the Jacobson radical. Our next concern are group rings, where we describe all commutative group rings, that are max. These are the group rings, that are composed from a commutative max ring and an abelian torsion group, where is finitely many elements of order pn for p not invertible in the ring. Finally we use this characterization to construct noncommutative group rings, which are max but not perfect.
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Group rings in coding theory
Horáček, Jan ; Žemlička, Jan (advisor) ; Šťovíček, Jan (referee)
This work is focused on linear error-correcting codes in group rings. The basic introduc- tion to group rings and to coding in group rings is given. By code we mean a R-submodul, which generalizes the definition of the code as an ideal. We describe unit-derived and zero- divisor codes. We test paramaters of unit-derived codes. The construction of LDPC codes without short cycles is explained. Except from the derivation of the generator and check matrix we focus on algebraic properties of codes and group rings. We also deal with the self-dual codes, reversible codes or number of units in a finite group algebra of a cyclic group. 1
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