Involutions in binary perfect codes

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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 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.
Original languageEnglish
Article number6006579
Pages (from-to)5926-5932
JournalIEEE Transactions on Information Theory
Volume57
Issue number9
DOIs
Publication statusPublished - 1 Sep 2011

Keywords

  • Automorphism group
  • involutions
  • perfect codes
  • rank
  • symmetry group

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