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

Duursma's reduced polynomial

Azniv Kasparian and  Ivan Marinov

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 ${\mathbb F}_q$.

Keywords: Homogeneous weight enumerator of a linear code, Duursma's zeta polynomial and Duursma's reduced polynomial of a linear code, Riemann Hypothesis Analogue for linear codes, formally self-dual linear codes, Hasse-Weil polynomial and Duursma's reduced polynomial of a function field of one variable.
Mathematics Subject Classification: Primary: 94B27, 14G50; Secondary: 11T71.
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.

[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.

[3]

I. Duursma, From weight enumerators to zeta functions, Discrete Applied Mathematics, 111 (2001), 55-73.  doi: 10.1016/S0166-218X(00)00344-9.

[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.

[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.

[6]

H. Niederreiter and Ch. Xing, Algebraic Geometry in Coding Theory and Cryptography Princeton University Press, 2009.

show all references

References:
[1]

S. Dodunekov and I. Landgev, Near MDS-codes, Journal of Geometry, 54 (1995), 30-43.  doi: 10.1007/BF01222850.

[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.

[3]

I. Duursma, From weight enumerators to zeta functions, Discrete Applied Mathematics, 111 (2001), 55-73.  doi: 10.1016/S0166-218X(00)00344-9.

[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.

[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.

[6]

H. Niederreiter and Ch. Xing, Algebraic Geometry in Coding Theory and Cryptography Princeton University Press, 2009.

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