Geodesic
Algebraic-geometry codes and decoding by error-correcting pairs
Prikladnaâ diskretnaâ matematika, no. 4 (2023), pp. 83-105

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We consider the basic theory of algebraic curves and their function fields necessary for constructing algebraic geometry codes and a pair of codes forming an error-correction pair which is used in a precomputation step of the decoding algorithm for the algebraic geometry codes. Also, we consider the decoding algorithm and give the necessary theory to prove its correctness. As a result, we consider elliptic curves, Hermitian curves and Klein quartics and construct the algebraic geometry codes associated with these families of curves, and also explicitly define the error-correcting pairs for the resulting codes.
Keywords: algebraic geometry code, function field, divisor, error-correcting pair, decoding of algebraic geometry code, elliptic curve, Hermitian curve
Mots-clés : Klein quartic. 
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     author = {E. S. Malygina and A. A. Kuninets and V. L. Ratochka and A. G. Duplenko and D. Y. Neyman},
     title = {Algebraic-geometry codes and decoding by error-correcting pairs},
     journal = {Prikladna\^a diskretna\^a matematika},
     pages = {83--105},
     publisher = {mathdoc},
     number = {4},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDM_2023_4_a6/}
}
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E. S. Malygina; A. A. Kuninets; V. L. Ratochka; A. G. Duplenko; D. Y. Neyman. Algebraic-geometry codes and decoding by error-correcting pairs. Prikladnaâ diskretnaâ matematika, no. 4 (2023), pp. 83-105. http://geodesic.mathdoc.fr/item/PDM_2023_4_a6/