There are lovely connections between certain characteristic 2 semifields and their associated translation planes and orthogonal spreads on the one hand, and Z 4 -linear Kerdock and Preparata codes on the other. These interrelationships 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 -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.

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Symplectic semifield planes and ℤ₄–linear codes

Publisher Summary
This chapter focuses on translation planes with emphasis on the classes of translation planes, dimensions and collineations, and collineation groups. A translation plane is an affine plane II whose translation group T (II) is (sharply) transitive on the affine points. (By a translation is meant a (P, l∞)-elation, where l∞ is the line at infinity, or improper line, of II and P ϵ l∞.) Every coordinatizing ternary ring of a translation plane (with respect to the line l∞ ) is quasified. A generalized Hall system is a quasifield derived from a semifield, and a generalized Hall plane is a translation plane coordinatized by a generalized Hall system. It the semifield is itself a field, that is the plane is Desarguesian, then the derived quasifield is called a Hall system, and the derived plane a Hall plane. A planar collineation of a translation plane II is a collineation which fixes a subplane pointwise. A planar collineation group of II is a collineation group which fixes a sub- plane pointwise.

Chapter 5 – Translation Planes

Translation planes of order q are constructed whose full collineation groups have order q2.

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Projective Planes of Order q whose Collineation Groups have Order q 2

Let ~r be a finite affine plane that permits derivation and let R denote a derivable net embedded in r. The collineation group of the derived plane r obtained by deriving R which leaves the replaced net/~ invariant induces a collineation group in r and which is called the inherited group. The main question with regard to the collineation group of is whether the full group of ~ is the inherited group.

The derivation of dual translation planes

There have been a number of striking new results concerning translation planes of characteristic 2, obtained using orthogonal and symplectic spreads. The impetus for this came from coding theory. This paper surveys the geometric advances, while providing a hint of their coding-theoretic connections.

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Chris Charnes

Papers

A non-symmetric translation plane of order 172