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Session S26 - Finite fields and applications

Thursday, July 15, 12:00 ~ 12:20 UTC-3

Bases of Riemann-Roch spaces from function fields of Kummer type and Algebraic Geometry Codes

Horacio Navarro

Universidad del Valle, Colombia   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

In 1981 Goppa defined the algebraic geometry codes (AG codes) as the image of Riemann-Roch spaces under the evaluation map at several rational places. It is important to note that the Riemann-Roch Theorem gives lower bounds for the dimension and minimum distance of this class of codes, however, to determine the dimension and a generator matrix it is necessary a basis of the associated Riemann-Roch space.

In 2005 Maharaj, Matthews and Pirsic constructed explicit bases of Riemann-Roch spaces related to the Hermitian function field based on its Kummer structure, in addition, they introduced the notion of the floor of a divisor in order to improve the bound of the minimum distance of the AG codes.

The purpose of this talk is to construct explicit bases of Riemann-Roch spaces associated to certain function fields of Kummer type, to calculate the floor of divisors with support on totally ramified places and finally to show examples of AG codes with good parameters.

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