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LetR′=GR(ps,psml)andR=GR(ps,psm)be two Galois rings. In this paper, we show how to construct normal basis in the extension of Galois rings, and we also define weakly self-dual normal basis and self-dual normal basis forR′overR, whereR′is considered as a free module overR. Moreover, we explain a way to construct self-dual normal basis using particular system of polynomials. Finally, we show the connection between self-dual normal basis forR′overRand the set of all invertible, circulant, and orthogonal matrices overR.

In this paper we study the structure of θ -cyclic codes over the ring B k including its connection to quasi-θ -cyclic codes over finite field F p r and skew polynomial rings over B k . We also characterize Euclidean self-dual θ -cyclic codes over the rings. Finally, we give the generator polynomial for such codes and some examples of optimal Euclidean θ -cyclic codes.

River water discharge is important information for water resources management planning, so it is necessary to develop river water discharge model as basis of its predictions. In order to get the result of predictions of river water discharge with high accuracy, it is developed a model of river water discharge based on the predictions of local climate (local rainfall and temperature) that are influenced by global climate conditions. Prediction of local climate is based on the Kernel nonparametric statistical downscaling model by utilizing GCM data. GCM data is a high dimensional global data, so data pre-processing is needed to reduce data dimension. It is done by CART algorithm. Statistical downscaling model is used to predict local rainfall and temperature. The prediction results are quite good with relatively small RMSE value. They are used to develop model of river water discharge. Modeling river water discharge is carried out using the Kernel nonparametric approach. The model of river water discharge produced is quite good because it can be used to predict river water discharge with relatively small RMSE.