Involutions in Binary Perfect Codes

Abstract

Given a 1-perfect code C, the group of symmetries of C, Sym(C)={π ∈ Sn | π(C)=C} , is a subgroup of the group of automorphisms of C. In this paper, we focus on symmetries of order two, i.e., involutions. Let InvF(C) ⊆ Sym(C) be the set of involutions that stabilize F pointwise. For linear 1-perfect codes, the possibilities for the number of fixed points |F| are given, establishing lower and upper bounds. For any m ≥ 2 and any value k between these bounds, [m/2] ≤ k ≤ m-1, linear 1-perfect codes of length n=2m-1 which have an involution that fixes |F| = 2k-1 coordinates are constructed. Moreover, for any m ≥ 4, 1 ≤ r ≤ m-1, and [m/2] ≤ k ≤ m-1, nonlinear 1-perfect codes of length n=2m-1 having rank n-m+r and an involution that fixes 2k-1 coordinates are also constructed, except one case, when m ≥ 6 is even, r=m-1 and k = [m/2].