A note on the symmetry group of full rank perfect binary codes
A perfect 1-error-correcting binary code is, for short, called a perfect code. The rank of a perfect code \(C\), \(\text{rank}(C)\), is the dimension of the linear span \(\langle C\rangle\) of the words of \(C\). It was proved by \textit{T. Etzion} and \textit{A. Vardy} [IEEE Trans. Inf. Theory 40, No. 3, 754--763 (1994; Zbl 0824.94029)] that for every length \(n= 2^m- 1\), where \(m\geq 4\), and every integer \(r\) in the interval \(n- \log(n+ 1)\leq r\leq n\) there exist perfect codes of rank \(r\) and length \(n\). A perfect code \(C\) has full rank if \(\text{rank}(C)= n\).NEWLINENEWLINE In this paper the author proves the following upper bound for the size of the symmetry groups of full rank perfect codes: For every full rank perfect code \(C\) of length \(n= 2^m-1\) with \(m\geq 4\), NEWLINE\[NEWLINE|\text{Sym}(C)|\leq{2\over n+1}|\text{GL}(\log(n+ 1),2)|,NEWLINE\]NEWLINE where \(\text{GL}(\log(n+ 1),2)\) means the general linear group. This upper bound is better than all previously known upper bounds.
scientific article; zbMATH DE number 6081343
scientific article