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 GL(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 Fqn of Fq.
Highlights
A Topological View of Reed–SolomonIn the last section as an application of the Horn problem, we provide a set of generators of the ideal associated with any algebraic code constructed on the normal curve (NRC) over an extension Fnq of Fq. on, Fq will be a field with q = pn elements and C a non-singular, projective, irreducible curve defined over Fq with q elements
We provide a set of generators for the algebraic code induced on the normal curve (NRC)
As we showed in Proposition 3, each subspace invariant under collineation of the NRC is indexed by a partition λ ∈ P (d)
Summary
In the last section as an application of the Horn problem, we provide a set of generators of the ideal associated with any algebraic code constructed on the NRC over an extension Fnq of Fq. on, Fq will be a field with q = pn elements and C a non-singular, projective, irreducible curve defined over Fq with q elements. A q−ary constant weight code of length n, distance d and weight w will be denoted as an [n, d, w]q code
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