Involutions in Binary Perfect Codes
Cristina Fernández-Córdoba, Kevin T. Phelps, Mercè Villanueva · IEEE Transactions on Information Theory · 2011
Given a 1-perfect codeC, the group of symmetries ofC,Sym(C)={π ∈Sn| π(C)=C} , is a subgroup of the group of automorphisms ofC. In this paper, we focus on symmetries of order two, i.e., involutions. LetInvF(C) ⊆Sym(C) be the set of involutions that stabilizeFpointwise. For linear 1-perfect codes, the possibilities for the number of fixed points |F| are given, establishing lower and upper bounds. For anym≥ 2 and any valuekbetween these bounds, [m/2] ≤k≤m-1, linear 1-perfect codes of lengthn=2m-1 which have an involution that fixes |F| = 2k-1 coordinates are constructed. Moreover, for anym≥ 4, 1 ≤r≤m-1, and [m/2] ≤k≤m-1, nonlinear 1-perfect codes of lengthn=2m-1 having rankn-m+rand an involution that fixes 2k-1 coordinates are also constructed, except one case, whenm≥ 6 is even,r=m-1 andk= [m/2].