A Topological View of Reed–Solomon Codes

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.

Geometry of a simplex inscribed in a normal rational curve

In 1959, Professor N. A. Court [2] generated synthetically a twisted cubic C circumscribing a tetrahedron T as the poles for T of the planes of a coaxal family whose axis is called the Lemoine axis of C for T. Here is an analytic attempt to relate a normal rational curve rn of order n, whose natural home is an n-space [n], with its Lemoine [n—2] L such that the first polars of points in L for a simplex S inscribed to rn pass through rn anf the last polars of points on rn for S pass through L. Incidently we come across a pair of mutually inscribed or Moebius simplexes but as a privilege of odd spaces only. In contrast, what happens in even spaces also presents a case, not less interesting, as considered here.

Self-polar double configurations in projective geometry

The principal theorem to be proved in this part is: Theorem II. If in IIn a normal rational curve, ρ, and a quadric primal S are such that there is a proper simplex inscribed in ρ and self-polar with regard to S, then there exist sets of N, = (2n+1/2), chords of р every two of which are conjugate with regard to S. A set can be constructed to contain any pair of chords of р which are conjugate with regard to S.