Discrete Fourier transforms (DFTs) over finite fields have widespread applications in digital communication and storage systems. Hence, reducing the computational complexities of DFTs is of great significance. Recently proposed cyclotomic fast Fourier transforms (CFFTs) are promising due to their low multiplicative complexities. Unfortunately, there are two issues with CFFTs: (1) they rely on efficient short cyclic convolution algorithms, which have not been sufficiently investigated in the literature and (2) they have very high additive complexities when directly implemented. To address both issues, we make three main contributions in this paper. First, for any odd prime , we reformulate a -point cyclic convolution as the product of a ( 1) ( 1) Toeplitz matrix and a vector, which has well-known efficient algorithms, leading to efficient bilinear algorithms for -point cyclic convolutions. Second, to address the high additive complexities of CFFTs, we propose composite cyclotomic Fourier transforms (CCFTs). In comparison to previously proposed fast Fourier transforms, our CCFTs achieve lower overall complexities for moderate to long lengths and the improvement significantly increases as the length grows. Third, our efficient algorithms for -point cyclic convolutions and CCFTs allow us to obtain longer DFTs over larger fields, e.g., the 2047-point DFT over GF(2 11 ) and 4095-point DFT over GF(2 12 ), which are first efficient DFTs of such lengths to the best of our knowledge. Finally, our CCFTs are also advantageous for hardware implementations due to their modular structure.
An automated path planning algorithm for a mobile robot in a structured environment is presented. An algorithm based on the Quine-McCluskey method of finding prime implicants in a logical expression is used to isolate all the largest rectangular free convex areas in a specified environment. The free convex areas are represented as nodes in a graph, and a graph traversal strategy that dynamically allocates costs to graph paths is used. Complexity of the algorithm and a strategy to trade optimality for smaller computation time are discussed.
We give a new algebraic representation for the wrap-around butterfly interconnection network. This new representation is based on the direct product of groups and finite fields and allows an algebraic expression of the network connectivity. The abstract algebraic tools may then be employed to explore the structural properties of the butterfly. In this paper we exploit this model to map guest graphs on the butterfly. In particular, we provide designs of unit dilation mappings of all possible length cycles on butterflies. We also map the largest possible binary trees on butterfly networks with a dilation 2 if the network degree is less than 16, 3 if it is less than 32, and 4 if it is less than 64. This is a great improvement over previous results.
The structure of bilinear cyclic convolution algorithms is explored over finite fields. The algorithms derived are valid for any length not divisible by the field characteristic and are based upon the small length polynomial multiplication algorithms. The multiplicative complexity of these algorithms is small and depends on the field of constants. The linear transformation matrices A, B (premultiplication), and C (postmultiplica-Manuscript
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