On the symmetry group of the Mollard code

We study the symmetry group of a binary perfect Mollard code $M(C,D)$ of length $tm+t+m$ containing as its subcodes the codes $C^1$ and $D^2$ formed from perfect codes $C$ and $D$ of lengths $t$ and $m$, respectively, by adding an appropriate number of zeros. For the Mollard codes, we generalize the result obtained in [1] for the symmetry group of Vasil'ev codes; namely, we describe the stabilizer $\mathrm{Stab}_{D^2}\mathrm{Sym}(M(C,D))$ of the subcode $D^2$ in the symmetry group of the code $M(C,D)$ (with the trivial function). Thus we obtain a new lower bound on the order of the symmetry group of the Mollard code. A similar result is established for the automorphism group of Steiner triple systems obtained by the Mollard construction but not necessarily associated with perfect codes. To obtain this result, we essentially use the notions of “linearity” of coordinate positions (points) of a nonlinear perfect code and a nonprojective Steiner triple system.