The Type K Gaussian Period Normal Basis And The Reed-Solomon Code Over Finite Fields

In this thesis, for the general type k Gaussian normal basisN, we obtain the dual basis and an upper bound for the complexity of N whenn≥k≥1. Furthermore, we prove that the upper bound can be achievedfor k = 3 and then determine all (weekly) self-dual type k Gaussian normalbases. Lastly, we get a necessary and su?cient condition for that there existsa generalized Reed-Solomon code which dual code is also a Reed-Solomon codeover finite fields. Furthermore, we construct a class of generalized Reed-Solomoncodes with the above property.