TY - JOUR
T1 - Involutions in binary perfect codes
AU - Fernández-Córdoba, Cristina
AU - Phelps, Kevin T.
AU - Villanueva, Mercè
PY - 2011/9/1
Y1 - 2011/9/1
N2 - 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 Inv F(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. © 2011 IEEE.
AB - 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 Inv F(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. © 2011 IEEE.
KW - Automorphism group
KW - involutions
KW - perfect codes
KW - rank
KW - symmetry group
U2 - 10.1109/TIT.2011.2162185
DO - 10.1109/TIT.2011.2162185
M3 - Article
SN - 0018-9448
VL - 57
SP - 5926
EP - 5932
JO - IEEE Transactions on Information Theory
JF - IEEE Transactions on Information Theory
IS - 9
M1 - 6006579
ER -