TRANSFORMATIONS OF CODES AND GEOMETRY RELATED TO VERONESEANS David G. Glynn Abstract. 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 9 3 configurations to another 9 3 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). 1. Introduction See [9, Chapter 1] for the following basic coding theory ideas. Definition 1.1. A linear code C with parameters [n, k, d] is the set of all vectors (codewords) of length n over GF (q) which are the linear combinations of the rows of a k × n “generator” matrix G over GF (q) of rank k. The weight of any codeword is just the number of its non-zero entries; in other words, the size of its support. The distance between any pair of codewords is the weight of the difference between these words. The parameter d is the minimum (non-zero) distance between distinct words in the code. Since the code is linear this is the same as the minimum weight of a non-zero word in the code. Definition 1.2. The linear code C has a dual code C ⊥ , which has an (n − k) × n generator matrix of rank n − k, often called parity check matrix, that is orthogonal to any generator matrix G of C with respect to the standard inner product. Thus C ⊥ is the set of all words of the same length n that are orthogonal to all words in the first code. If the first code has parameters [n, k, d], then the second has parameters [n, n − k,d ′ ], and a parity check matrix of C is a generator matrix of C ⊥ and vice-versa. 1991 Mathematics Subject Classification. 05E20 14M99 51E20 51M35 94B60 05B07 05B25 05B35. Key words and phrases. code, projective plane, hyperoval, functions, duality, Veronesean, information set. Typeset by A M S-T E X 1