The ring Fp + uFp + + uk-1Fp may be of interest in coding theory, which have already been used in the construction of optimal frequency-hopping sequence. In this work, cyclic codes over Fp + uFp + + uk-1Fp which is an open problem posed in [1] are considered. Namely, the structure of cyclic code over Fp + uFp + + uk-1Fp and that of their duals are derived.

Mining generalized frequent patterns of generalized association rules is an important process in knowledge discovery system. In this paper, we propose a new approach for efficiently mining all frequent patterns using a novel set enumeration algorithm with two types of constraints on two generalized itemset relationships, called subset-superset and ancestor-descendant constraints. We also show a method to mine a smaller set of generalized closed frequent itemsets instead of mining a large set of conventional generalized frequent itemsets. To this end, we develop two algorithms called SET and cSET for mining generalized frequent itemsets and generalized closed frequent itemsets, respectively. By a number of experiments, the proposed algorithms outperform the previous well-known algorithms in both computational time and memory utilization. Furthermore, the experiments with real datasets indicate that mining generalized closed frequent itemsets gains more merit on computational costs since the number of generalized closed frequent itemsets is much more smaller than the number of generalized frequent itemsets.

In this paper, we present a polynomial construction based on Jacket and Hadamard matrices over the Galois Field. The construction has two modes, one only includes matrices extension, and the other includes a center-weighted scheme for polynomial representations. Here, an "addition" scheme is used to represent matrices, which can lead to simple operations and convenient implementation of hardware.

This paper presents a novel and successful optimization algorithm for optimizing Multiple-valued Logic (MVL) functions based on Petri net theory. Mathematical properties and Petri net modeling tools to implement MVL systems are introduced. On the basis of these properties and modeling tools, the optimization algorithm can synthesize, analyze and minimize an arbitrary quaternary logic function of n-input variables. The analysis technique of optimization algorithm is a well-established concept from both theories of MVL and Petri nets, and this can be applied to specify and optimize any MVL Petri net system. In this paper, Petri nets of Galois field have been proposed in order to form a complete system, which can be used to realize and construct VLSI circuit of any MVL function. Based on the Petri nets of Galois field and the proposed algorithm, the quaternary minimum and maximum functions have been analyzed, minimized, and designed. These applications have demonstrated the usefulness of optimization algorithm. Based on Petri net theory, the analysis revealed important information about MVL Petri net modeled systems, where this information has been used to evaluate the modeled system and suggest improvements or changes. For evaluation, advantages of the proposed method over a conventional logic minimization method are presented. Also, we have observed that the MVL Petri nets have the following advantages: Designers can exhibit clearly, simply and systematically any complex MVL Petri net nodel, number of concurrent operations is increased, number of places and transitions that are needed to realize a MVL model is very small, and the interconnection problems can be greatly reduced.