One-Point Algebraic Geometric Codes from Artin-Schreier Extensions of Hermitian Function Fields

Daisuke UMEHARA; Tomohiko UYEMATSU

One-Point Algebraic Geometric Codes from Artin-Schreier Extensions of Hermitian Function Fields

Daisuke UMEHARA, Tomohiko UYEMATSU

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Recently, Garcia and Stichtenoth proposed sequences of algebraic function fields with finite constant fields such that their sequences attain the Drinfeld-Vl bound. In the sequences, the third algebraic function fields are Artin-Schreier extensions of Hermitian function fields. On the other hand, Miura presented powerful tools to construct one-point algebraic geometric (AG) codes from algebraic function fields. In this paper, we clarify rational functions of the third algebraic function fields which correspond to generators of semigroup of nongaps at a specific place of degree one. Consequently, we show generator matrices of the one-point AG codes with respect to the third algebraic function fields for any dimension by using rational functions of monomial type and rational points.

Publication: IEICE TRANSACTIONS on Fundamentals Vol.E81-A No.10 pp.2025-2031

Publication Date: 1998/10/25

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Type of Manuscript: Special Section PAPER (Special Section on Information Theory and Its Applications)

Category: Coding Theory

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Daisuke UMEHARA, Tomohiko UYEMATSU, "One-Point Algebraic Geometric Codes from Artin-Schreier Extensions of Hermitian Function Fields" in IEICE TRANSACTIONS on Fundamentals, vol. E81-A, no. 10, pp. 2025-2031, October 1998, doi: .
Abstract: Recently, Garcia and Stichtenoth proposed sequences of algebraic function fields with finite constant fields such that their sequences attain the Drinfeld-Vl bound. In the sequences, the third algebraic function fields are Artin-Schreier extensions of Hermitian function fields. On the other hand, Miura presented powerful tools to construct one-point algebraic geometric (AG) codes from algebraic function fields. In this paper, we clarify rational functions of the third algebraic function fields which correspond to generators of semigroup of nongaps at a specific place of degree one. Consequently, we show generator matrices of the one-point AG codes with respect to the third algebraic function fields for any dimension by using rational functions of monomial type and rational points.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/e81-a_10_2025/_p

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@ARTICLE{e81-a_10_2025,
author={Daisuke UMEHARA, Tomohiko UYEMATSU, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={One-Point Algebraic Geometric Codes from Artin-Schreier Extensions of Hermitian Function Fields},
year={1998},
volume={E81-A},
number={10},
pages={2025-2031},
abstract={Recently, Garcia and Stichtenoth proposed sequences of algebraic function fields with finite constant fields such that their sequences attain the Drinfeld-Vl bound. In the sequences, the third algebraic function fields are Artin-Schreier extensions of Hermitian function fields. On the other hand, Miura presented powerful tools to construct one-point algebraic geometric (AG) codes from algebraic function fields. In this paper, we clarify rational functions of the third algebraic function fields which correspond to generators of semigroup of nongaps at a specific place of degree one. Consequently, we show generator matrices of the one-point AG codes with respect to the third algebraic function fields for any dimension by using rational functions of monomial type and rational points.},
keywords={},
doi={},
ISSN={},
month={October},}

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TY - JOUR
TI - One-Point Algebraic Geometric Codes from Artin-Schreier Extensions of Hermitian Function Fields
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 2025
EP - 2031
AU - Daisuke UMEHARA
AU - Tomohiko UYEMATSU
PY - 1998
DO -
JO - IEICE TRANSACTIONS on Fundamentals
SN -
VL - E81-A
IS - 10
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - October 1998
AB - Recently, Garcia and Stichtenoth proposed sequences of algebraic function fields with finite constant fields such that their sequences attain the Drinfeld-Vl bound. In the sequences, the third algebraic function fields are Artin-Schreier extensions of Hermitian function fields. On the other hand, Miura presented powerful tools to construct one-point algebraic geometric (AG) codes from algebraic function fields. In this paper, we clarify rational functions of the third algebraic function fields which correspond to generators of semigroup of nongaps at a specific place of degree one. Consequently, we show generator matrices of the one-point AG codes with respect to the third algebraic function fields for any dimension by using rational functions of monomial type and rational points.
ER -