Counting polynomials over finite fields with prescribed leading coefficients and linear factors |
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| Affiliation: | 1. University of Rijeka, Faculty of Mathematics, Radmile Matejčić 2, 51000 Rijeka, Croatia;2. Ghent University, Department of Mathematics: Algebra and Geometry, Gent, Flanders, Belgium;3. University of Canterbury, School of Mathematics and Statistics, Private Bag 4800, 8140 Christchurch, New Zealand;1. Departamento de Matemática, Facultad de Ciencias Económicas, UNL, Moreno 2557, 3000 Santa Fe, Argentina;2. Researcher of CONICET at FIQ, Universidad Nacional del Litoral, Santiago del Estero 2829, 3000 Santa Fe, Argentina;3. FaMAF-CIEM (CONICET), Universidad Nacional de Córdoba, Av. Medina Allende 2144, Ciudad Universitaria, 5000 Córdoba, Argentina;4. Departamento de Matemática, Facultad de Ingeniería Química, UNL, Santiago del Estero 2829, 3000 Santa Fe, Argentina;1. Department of Mathematics, Niigata University, Niigata 950-2181, Japan;2. Graduate School of Science and Technology, Niigata University, Niigata 950-2181, Japan;1. College of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China;2. College of Information Science and Technology, Jinan University, Guangzhou 510632, China;3. National Center for Applied Mathematics in Hunan, Xiangtan University, Xiangtan 411105, China |
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| Abstract: | ![]()
We count the number of polynomials over finite fields with prescribed leading coefficients and a given number of linear factors. This is equivalent to counting codewords in Reed-Solomon codes which are at a certain distance from a received word. We first apply the generating function approach, which is recently developed by the author and collaborators, to derive expressions for the number of monic polynomials with prescribed leading coefficients and linear factors. We then apply Li and Wan's sieve formula to simplify the expressions in some special cases. Our results extend and improve some recent results by Li and Wan, and Zhou, Wang and Wang. |
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| Keywords: | Finite fields Counting Polynomials Prescribed coefficients Reed-Solomon codes Distance distribution |
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