Rigidity Theorems for a Class of Affine Resolvable Designs

Abstract

The affine resolvable 2-(27,9,4) designs were classified by Lam and Tonchev [9, 10]. We use their construction of the designs to examine the ternary codes of the designs and show, using Magma [3], that each of the codes, apart from two, contains, amongst its constant weight-9 codewords, a copy of the ternary code of the affine geometry design of points and planes in $A{G}_3(F_3)$. We also show how the ternary codes of the 68 designs and of their dual designs, together with properties of the automorphism groups of the designs, can be used to characterize the designs.