INTRODUCTION

The amount of information which coding theory gives about a design can vary enormously from one design to another. As it is now known that a plane of order 10 can have only the identical collineation, its binary code is now our main source of information on it.

Section 2 gives the definition of the code of a design in terms of its incidence matrix.

For a (v, b, r, k, λ) design D it is advantageous to consider codes over finite fields Fq, where q = pt, p a prime dividing r - λ, and in Section 3 theorems dealing with this are given. The Thompson-Hayden theorems of this section deal with a vector space V over a field F and the action of a finite orthogonal group G whose order has an inverse in F. We are concerned with the action of G on a subspace C of V which is a G-module. This is particularly appropriate when C is a code and G a group of automorphisms of C.

In Section 4 the role of coding theory is dicussed in connection with the plane of order 10 and the construction of a (41, 16, 6) design.

THE CODE OF A DESIGN

A balanced incomplete block design D, or more briefly design, is a system of v points a1,…,av and b blocks B1,…Bb, together with an incidence relation ai ∈ Bj (read ai belongs to Bj or Bj contains ai) between certain points ai and blocks Bj.