Symplectic semifield planes and ℤ₄–linear codes

Abstract

There are lovely connections between certain characteristic 2 semifields and their associated translation planes and orthogonal spreads on the one hand, and Z 4 \mathbb {Z}_4 –linear Kerdock and Preparata codes on the other. These inter–relationships lead to the construction of large numbers of objects of each type. In the geometric context we construct and study large numbers of nonisomorphic affine planes coordinatized by semifields; or, equivalently, large numbers of non–isotopic semifields: their numbers are not bounded above by any polynomial in the order of the plane. In the coding theory context we construct and study large numbers of Z 4 \mathbb {Z}_4 –linear Kerdock and Preparata codes. All of these are obtained using large numbers of orthogonal spreads of orthogonal spaces of maximal Witt index over finite fields of characteristic 2. We also obtain large numbers of “boring” affine planes in the sense that the full collineation group fixes the line at infinity pointwise, as well as large numbers of Kerdock codes “boring” in the sense that each has as small an automorphism group as possible. The connection with affine planes is a crucial tool used to prove inequivalence theorems concerning the orthogonal spreads and associated codes, and also to determine their full automorphism groups.