Involutions in Additive 1-Perfect Codes

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 2, i.e., involutions. Let Inv(C) ⊆ Sym(C) be the set of involutions of C. An additive code C of type (α,β) is a subgroup of BBZ2^α×BBZ4^β. For additive 1-perfect codes C of length 2^t-1, we establish that any involution π ∈ Inv(C) must have at least 2^k-1 fixed points where k ≥ [t/2].