Perfect codes in $SL (2,2^f)$ (Algebraic Combinatorics)
Sachiyo Terada · Institutional Repositories DataBase (IRDB) · 2003
We show that the Cayley graph $\Gamma(SL(2,2^{f})$ , $X)$ of the fi- nite special linear group $SL(2,2^{f})$ does not have any perfect code if $X$ is closed under conjugation for anatural integer $f\geq 2$ .Moreover, as acase where $X$ is not closed under conjugation, we consider the orbits $X$ of involutions by conjugation of aSinger cycle of $SL(2,2^{f})$ and determine whether they divide $\lambda SL(2,2^{f})$ non-trivially or not.1IntroductionWe study acombinatorial problem below in the finite special linear groups $SL(2,2^{f})$ .Problem.Determine the existence of perfect codes in aCay- ley graph.Perfect codes have been mainly studied over finite fields.Re- cently perfect codes are studied in distance-transitive graphs and distance-regular graphs.As acase of agraph which is not