On Pless symmetry codes, ternary QR codes, and related Hadamard matrices and designs

It is proved that a code L(q) which monomially equivalent to the Pless symmetry C(q) of length $$2q+2$$ contains (0,1)-incidence matrix Hadamard 3- $$(2q+2,q+1,(q-1)/2)$$ design D(q) associated with Paley–Hadamard type II. Similarly, any ternary extended quadratic residue incidence 3-design I. If $$q=5, 11, 17, 23$$ , then full permutation automorphism group coincides D(q), and similar result holds for codes lengths 24 48. All matrices order 36 formed by codewords C(17) are enumerated classified up equivalence. There two equivalence classes such matrices: H I 19584, second regular $$H'$$ symmetric 2-(36, 15, 6) D has trivial group, spans C(17).