On the representations in GF(3)^4 of the Hadamard design H_11

In this paper we study the representations of the 2-(11,5,2) Hadamard design H_11 = (P,B) as a set of eleven points in the 4-dimensional vector space GF(3)^4, under the conditions that the five points in each block sum up to zero, and dim ‹P› = 4. We show that, up to linear automorphism, there exist precisely two distinct, linearly nonisomorphic representations, and, in either case, we characterize the family S of all the 5-subsets of P whose elements sum up to zero. In both cases, S properly contains the family of blocks B, thereby showing that a previous result on the representations of H_11 in GF(3)^5 cannot be improved.