A Topological View of Reed–Solomon Codes

Alberto Besana and Cristina Martínez
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Alberto Besana: Department of Physics and Mathematics, University of Alcalá, 28871 Madrid, Spain
Cristina Martínez: Department of Physics and Mathematics, University of Alcalá, 28871 Madrid, Spain

Mathematics, 2021, vol. 9, issue 5, 1-16

Abstract: We studied a particular class of well known error-correcting codes known as Reed–Solomon codes. We constructed RS codes as algebraic-geometric codes from the normal rational curve. This approach allowed us to study some algebraic representations of RS codes through the study of the general linear group G L ( n , q ) . We characterized the coefficients that appear in the decompostion of an irreducible representation of the special linear group in terms of Gromov–Witten invariants of the Hilbert scheme of points in the plane. In addition, we classified all the algebraic codes defined over the normal rational curve, thereby providing an algorithm to compute a set of generators of the ideal associated with any algebraic code constructed on the rational normal curve (NRC) over an extension F q n of F q .

Keywords: 2000 mathematics subject classification; 05E10 (primary); 05A15 (secondary); algebraic code; symmetric group; partitions (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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