2-designs overGF(2^m)

Abstract

At − (n, k, λ; q) design is a collection ℬ ofk-dimensional subspaces of ann-dimensional vector space overGF(q) with the property that anyt-dimensional subspace is contained in exactlyλ members of ℬ. It is also called a design over a finite field or aq-analoguet-design. The first nontrivial example fort ≥ 2 was given by S. Thomas. Namely, he constructed a series of 2 − (n, 3, 7; 2) design for alln ≥ 7 satisfying (n, 6) = 1. Under the same restriction onn, we show that the base field of Thomas' design is extensible toGF(2^m), i.e., we construct a 2 − (n, 3, 2^2m + 2^m + 1; 2^m) design for allm ≥ 1.