Performance of codes based on crossed product algebras

Abstract

The work presented in this thesis is concerned with algebraic coding theory, with a particular focus on space-time codes constructed from crossed product algebras. This thesis is divided into three parts. In the first part we will present a method for constructing codes from crossed product algebras and derive bounds on their performance. The second part concerns itself with codes constructed from cyclic algebras. Finally in the third part, constructions based on biquadratic crossed product algebras are considered. It is well known that two important design criteria in the construction of spacetime codes are the rank criterion and the determinant criterion. The rank criterion is closely linked to the notion of fully diverse codes. Constructing codes that are fully diverse led to the study of codes based on division algebras. To give explicit constructions of codes, central simple algebras were considered and in particular crossed product algebras. In this thesis we derive bounds on the minimum determinant of codes constructed from crossed product algebras. A lot of work has focused on constructing codes based on cyclic division algebras. The well known perfect space-time block codes are codes that satisfy a variety of coding constraints that make them very efficient for coding. We consider the performance of these codes and prove that the best known examples are optimal with respect to the coding gain. Finally we consider codes based on biquadratic crossed product algebras, where the Galois group of the underlying field extension is isomorphic to the Klein fourgroup. It has been shown that these codes can satisfy a large number of coding criteria and exhibit very good performance. We prove the optimality of the best known code

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