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**1 - 1**of**1**### TRANSFORMATIONS OF CODES AND GEOMETRY RELATED TO VERONESEANS

"... There is a chain of polynomial codes that contains the simplex code of the projective plane over GF (q). It is related to Veroneseans of the plane. We show how to construct information sets for some of these codes using any dual hyperoval in such a plane. Also, the more general Veroneseans of hype ..."

Abstract
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There is a chain of polynomial codes that contains the simplex code of the projective plane over GF (q). It is related to Veroneseans of the plane. We show how to construct information sets for some of these codes using any dual hyperoval in such a plane. Also, the more general Veroneseans of hypersurfaces of degree i of projective space are considered and, related to this, a general transformation of codes and of sets of points in projective geometry that generalizes coding theoretic duality. We call it “duality of order i”. If we ensure that a set of points is taken to another set of points in the same space then the transformation is invertible for generic sets of points. First order duality corresponds to the usual duality of codes and of matroids. Quadratic duality takes any 93 configurations to another 93 e.g. Pappus. One of the Steiner triple systems having 13 points is taken to the projective plane PG(2, 3) of order 3 using an abstract version of the third order duality. This leads to a construction of the triple system using 26 conics in PG(2, 3).