A note on regular subgroups of the automorphism group of the linear Hadamard code

We consider the subgroups of the automorphism group of the linear Hadamard code that act regularly on the codewords of the code. These subgroups correspond to the regular subgroups of the general affine group $\mathrm{GA}(r,2)$ with respect to the action on the vectors of $F_2^{r}$, where $n=2^r-1 $ is the length of the Hamadard code. We show that the dihedral group $D_{2^{r-1}}$ is a regular subgroup of $\mathrm{GA}(r,2)$ only when $r=3$. Following the approach of [13] we study the regular subgroups of the automorphism group of the Hamming code obtained from the regular subgroups of the automorphism group of the Hadamard code of length $15$.