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14-09-2022 | Original Paper

Note on the dimension of Goppa codes

Authors: Xiaoshan Quan, Qin Yue

Published in: Applicable Algebra in Engineering, Communication and Computing | Issue 5/2024

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Abstract

Let \(\Gamma (L, g)\) be a Goppa code over \({\mathbb {F}}_q\), where \(L\subset \mathbb {F}_{q^{m}}\) is a support and \(g(x)\in \mathbb {F}_{q^{m}}[x]\) is a polynomial with s distinct roots in \({\mathbb {F}}_{q^m}\). In [Couvreur A, Otmani A, Tillich JP (2014) New identities relating wild Goppa codes. Finite Field Appl 29: 178–197.], Couvreur at al. gave the bound: \(\dim _{{\mathbb {F}}_{q}}\Gamma (L,g^e)-\dim _{{\mathbb {F}}_{q}}\Gamma (L,g^{e+1})\le s,\) where \(e=q^{m-1}+q^{m-2}+\cdots +q\). In this paper, we give the conditions such that \(\dim _{{\mathbb {F}}_{q}}\Gamma (L,g^e)=\dim _{{\mathbb {F}}_{q}}\Gamma (L,g^{e+1})\).

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Metadata
Title
Note on the dimension of Goppa codes
Authors
Xiaoshan Quan
Qin Yue
Publication date
14-09-2022
Publisher
Springer Berlin Heidelberg
Published in
Applicable Algebra in Engineering, Communication and Computing / Issue 5/2024
Print ISSN: 0938-1279
Electronic ISSN: 1432-0622
DOI
https://doi.org/10.1007/s00200-022-00578-z

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