Decreasing norm-trace codes

The decreasing norm-trace codes are evaluation codes defined by a set of monomials closed under divisibility and the rational points of the extended norm-trace curve. In particular, the decreasing norm-trace codes contain the one-point algebraic geometry (AG) codes over the extended norm-trace curve...

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Bibliographic Details
Published inDesigns, codes, and cryptography Vol. 92; no. 5; pp. 1143 - 1161
Main Authors Carvalho, Cícero, López, Hiram H., Matthews, Gretchen L.
Format Journal Article
LanguageEnglish
Published New York Springer US 01.05.2024
Springer Nature B.V
Subjects
Coding and Information Theory
Computer Science
Cryptology
Discrete Mathematics in Computer Science
Mathematical analysis
Norm-trace curves
11T71
14G50
Evaluation codes
94B05
Repair scheme
Decreasing monomial codes
Dual codes
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Summary:The decreasing norm-trace codes are evaluation codes defined by a set of monomials closed under divisibility and the rational points of the extended norm-trace curve. In particular, the decreasing norm-trace codes contain the one-point algebraic geometry (AG) codes over the extended norm-trace curve. We use Gröbner basis theory and find the indicator functions on the rational points of the curve to determine the basic parameters of the decreasing norm-trace codes: length, dimension, and minimum distance. We also obtain their dual codes. We give conditions for a decreasing norm-trace code to be a self-orthogonal or a self-dual code. We provide a linear exact repair scheme to correct single erasures for decreasing norm-trace codes, which applies to higher rate codes than the scheme developed by Jin et al. (IEEE Trans Inf Theory 64(2):900–908, 2018) when applied to the one-point AG codes over the extended norm-trace curve.
ISSN:0925-1022
1573-7586
DOI:10.1007/s10623-023-01334-1