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Applicable Algebra in Engineering, Communication and Computing

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01-11-2013 | Original Paper

On the dual minimum distance and minimum weight of codes from a quotient of the Hermitian curve

Authors: Edoardo Ballico, Alberto Ravagnani

Published in: Applicable Algebra in Engineering, Communication and Computing | Issue 5/2013

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Abstract

In this paper we study evaluation codes arising from plane quotients of the Hermitian curve, defined by affine equations of the form \(y^q+y=x^m,\,q\) being a prime power and \(m\) a positive integer which divides \(q+1\). The dual minimum distance and minimum weight of such codes are studied from a geometric point of view. In many cases we completely describe the minimum-weight codewords of their dual codes through a geometric characterization of the supports, and provide their number. Finally, we apply our results to describe Goppa codes of classical interest on such curves.

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Here the subscript red denotes the reduction of a zero-dimensional scheme.

We recall that \(\text{ Res }_L (A \cup E) = \text{ Res }_L (E)\), because \(A\subseteq L\), and \(\deg (\text{ Res }_L (E)) = \deg (E) -\deg (L \cap E)\).

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Title: On the dual minimum distance and minimum weight of codes from a quotient of the Hermitian curve
Authors: Edoardo Ballico
Alberto Ravagnani
Publication date: 01-11-2013
Publisher: Springer Berlin Heidelberg
Published in: Applicable Algebra in Engineering, Communication and Computing / Issue 5/2013
Print ISSN: 0938-1279
Electronic ISSN: 1432-0622
DOI: https://doi.org/10.1007/s00200-013-0206-z

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