On the Symmetry Group of Perfect 1-Error Correcting Binary Codes

It is shown that for any rank \( r \) with \( n – \log(n+1) + 4 \leq r \leq n – 4 \) and any length \( n \), where \( n = 2^k – 1 \) and \( k \geq 8 \), there is a perfect code with these parameters and with a trivial group of symmetries.