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In this note we report the classification of all symmetric 2-(36,15,6)designs that admit an automorphism of order 2 and their incidence matricesgenerate an extremal ternary self-dual code. It is shown that up toisomorphism, there exists only one such design, having a full automorphismgroup of order 24, and the ternary code spanned by its incidence matrix isequivalent to the Pless symmetry code.

Extremal ternary self-dual codes of length 36 and symmetric 2-(36,15,6) designs with an automorphism of order 2