# American Institute of Mathematical Sciences

November  2017, 11(4): 647-669. doi: 10.3934/amc.2017048

## Duursma's reduced polynomial

 Section of Algebra, Department of Mathematics and Informatics, Kliment Ohridski University of Sofia, James Bouchier Blvd., Sofia 1164, Bulgaria

Received  August 2014 Published  November 2017

Fund Project: Supported by Contract 015/9.04.2014 with the Scientific Foundation of the University of Sofia.

The weight distribution $\{ \mathcal{W}_C^{(w)} \} _{w=0} ^n$ of a linear code $C \subset {\mathbb F}_q^n$ is put in an explicit bijective correspondence with Duursma's reduced polynomial $D_C(t) ∈ {\mathbb Q}[t]$ of $C$. We prove that the Riemann Hypothesis Analogue for a linear code $C$ requires the formal self-duality of $C$. Duursma's reduced polynomial $D_F(t) ∈ {\mathbb Z}[t]$ of the function field $F = {\mathbb F}_q(X)$ of a curve $X$ of genus $g$ over ${\mathbb F}_q$ is shown to provide a generating function $\frac{D_F(t)}{(1-t)(1-qt)} = \sum\limits _{i=0} ^{∞} \mathcal{B}_i t^{i}$ for the numbers $\mathcal{B}_i$ of the effective divisors of degree $i ≥0$ of a virtual function field of a curve of genus $g-1$ over ${\mathbb F}_q$.

Citation: Azniv Kasparian, Ivan Marinov. Duursma's reduced polynomial. Advances in Mathematics of Communications, 2017, 11 (4) : 647-669. doi: 10.3934/amc.2017048
##### References:
 [1] S. Dodunekov and I. Landgev, Near MDS-codes, Journal of Geometry, 54 (1995), 30-43.  doi: 10.1007/BF01222850.  Google Scholar [2] I. Duursma, Weight distribution of geometric Goppa codes, Transections of the American Mathematical Society, 351 (1999), 3609-3639.  doi: 10.1090/S0002-9947-99-02179-0.  Google Scholar [3] I. Duursma, From weight enumerators to zeta functions, Discrete Applied Mathematics, 111 (2001), 55-73.  doi: 10.1016/S0166-218X(00)00344-9.  Google Scholar [4] I. Duursma, Combinatorics of the two-variable zeta function in Finite Fields and Applications, Lecture Notes in Computational Sciences, Springer, Berlin, 2948 (2004), 109-136.  Google Scholar [5] D. Ch. Kim and J. Y. Hyun, A Riemann hypothesis analogue for near-MDS codes, Discrete Applied Mathematics, 160 (2012), 2440-2444.  doi: 10.1016/j.dam.2012.07.008.  Google Scholar [6] H. Niederreiter and Ch. Xing, Algebraic Geometry in Coding Theory and Cryptography Princeton University Press, 2009.  Google Scholar

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##### References:
 [1] S. Dodunekov and I. Landgev, Near MDS-codes, Journal of Geometry, 54 (1995), 30-43.  doi: 10.1007/BF01222850.  Google Scholar [2] I. Duursma, Weight distribution of geometric Goppa codes, Transections of the American Mathematical Society, 351 (1999), 3609-3639.  doi: 10.1090/S0002-9947-99-02179-0.  Google Scholar [3] I. Duursma, From weight enumerators to zeta functions, Discrete Applied Mathematics, 111 (2001), 55-73.  doi: 10.1016/S0166-218X(00)00344-9.  Google Scholar [4] I. Duursma, Combinatorics of the two-variable zeta function in Finite Fields and Applications, Lecture Notes in Computational Sciences, Springer, Berlin, 2948 (2004), 109-136.  Google Scholar [5] D. Ch. Kim and J. Y. Hyun, A Riemann hypothesis analogue for near-MDS codes, Discrete Applied Mathematics, 160 (2012), 2440-2444.  doi: 10.1016/j.dam.2012.07.008.  Google Scholar [6] H. Niederreiter and Ch. Xing, Algebraic Geometry in Coding Theory and Cryptography Princeton University Press, 2009.  Google Scholar
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