Involutions in Binary 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 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
=2
m
-1 which have an involution that fixes |
F
| = 2
k
-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
=2
m
-1 having rank
n
-
m
+
r
and an involution that fixes 2
k
-1 coordinates are also constructed, except one case, when
m
≥ 6 is even,
r
=
m
-1 and
k
= [
m
/2].