Abstract
This paper provides a fast algorithm for Gröbner bases of homogenous ideals of \( \mathbb{F} \)[x, y] over a finite field \( \mathbb{F} \). We show that only the S-polynomials of neighbor pairs of a strictly ordered finite homogenours generating set are needed in the computing of a Gröbner base of the homogenous ideal. It reduces dramatically the number of unnecessary S-polynomials that are processed. We also show that the computational complexity of our new algorithm is O(N 2), where N is the maximum degree of the input generating polynomials. The new algorithm can be used to solve a problem of blind recognition of convolutional codes. This problem is a new generalization of the important problem of synthesis of a linear recurring sequence.
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Supported by the National Natural Science Foundation of China (Grant No. 60673082), Special Funds of Authors of Excellent Doctoral Dissertation in Chian (Grant No. 200084)
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Lu, P., Zou, Y. Fast computation of Gröbner basis of homogenous ideals of \( \mathbb{F} \)[x, y]. Sci. China Ser. F-Inf. Sci. 51, 368–380 (2008). https://doi.org/10.1007/s11432-008-0032-2
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DOI: https://doi.org/10.1007/s11432-008-0032-2