Perfect codes in \(\mathrm{SL}(2,2^f)\) (Q1883617)

For a non-empty subset \(X\) of a finite group \(G\) and a positive integer \(\lambda\), \(X\) is said to divide \(\lambda G\) if there is a subset \(Y\) of \(G\) such that each element \(g\) of \(G\) can be represented as \(g= xy\) with \(x\in X\) and \(y\in Y\) in exactly \(\lambda\) ways. The subset \(Y\) is called a code with respect to \(X\). It is shown that there exists no perfect code in the Cayley graph of \(\mathrm{SL}(2,2^f)\) with respect to a subset \(X\) with \(f\neq 1\) which is closed under conjugation. A list of subsets \(X\) closed under conjugation and positive integers \(\lambda\) such that \(X\) possibly divides \(\lambda\mathrm{SL}(2, 2^f)\) has been established.