Stopping Sets of Algebraic Geometry Codes over Hyperelliptic Curves of Genus Two

Eid, Abdulla

doi:10.3390/math12223522

Open AccessArticle

Stopping Sets of Algebraic Geometry Codes over Hyperelliptic Curves of Genus Two

by

Abdulla Eid

Department of Mathematics, University of Bahrain, Sakhir Campus, Zallaq P.O. Box 32038, Bahrain

Mathematics 2024, 12(22), 3522; https://doi.org/10.3390/math12223522

Submission received: 29 September 2024 / Revised: 8 November 2024 / Accepted: 9 November 2024 / Published: 12 November 2024

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Abstract

Stopping sets are useful for analyzing the performance of a linear code under an iterative decoding algorithm over an erasure channel. In this paper, we consider stopping sets of one-point algebraic geometry codes defined by a hyperelliptic curve of genus

g = 2

defined by the plane model

y^{2} = f (x)

, where the degree of

f (x)

was 5. We completely classify the stopping sets of the one-point algebraic geometric codes

C = C_{Ω} (D, m P_{\infty})

defined by a hyperelliptic curve of genus 2 with

m \leq 4

. For

m = 3

, we proved in detail that all sets

S \subseteq {1, 2, \dots, n}

of a size greater than 3 are stopping sets and we give an example of sets of size

2, 3

that are not.

Keywords: stopping sets; algebraic geometric codes; Riemann–Roch spaces; hyperelliptic curves

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MDPI and ACS Style

Eid, A. Stopping Sets of Algebraic Geometry Codes over Hyperelliptic Curves of Genus Two. Mathematics 2024, 12, 3522. https://doi.org/10.3390/math12223522

AMA Style

Eid A. Stopping Sets of Algebraic Geometry Codes over Hyperelliptic Curves of Genus Two. Mathematics. 2024; 12(22):3522. https://doi.org/10.3390/math12223522

Chicago/Turabian Style

Eid, Abdulla. 2024. "Stopping Sets of Algebraic Geometry Codes over Hyperelliptic Curves of Genus Two" Mathematics 12, no. 22: 3522. https://doi.org/10.3390/math12223522

APA Style

Eid, A. (2024). Stopping Sets of Algebraic Geometry Codes over Hyperelliptic Curves of Genus Two. Mathematics, 12(22), 3522. https://doi.org/10.3390/math12223522

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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Stopping Sets of Algebraic Geometry Codes over Hyperelliptic Curves of Genus Two

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