A note on the symmetry group of full rank perfect binary codes

On the Automorphism Groups of the $$\mathbb{Z}_{2}\mathbb{Z}_{4}$$ -Linear Hadamard Codes and Their Classification

“…The permutation automorphism group of Z 2 Z 4 -linear (extended) 1-perfect codes has been studied in [15,19]. The permutation automorphism group of (nonlinear) binary 1-perfect codes has also been studied before, obtaining some partial results [1,[9][10][11]. Finally, the permutation automorphism group of Z 2 Z 4 -additive Hadamard codes has been studied in [18].…”

On the symmetry group of the Mollard code

“…The order of the automorphism group of an arbitrary nonlinear perfect binary code was investigated by several authors, see the papers [19,20,11,6,7]. The main definitions concerning this paper see in [9].…”

“…It is well known [9] that the symmetry group Sym(H) of the Hamming code H of length n is isomorphic to the general linear group GL(log(n + 1), 2). By the linearity of the Hamming code H of length n, we have The order of the automorphism group of an arbitrary nonlinear perfect binary code was investigated by several authors, see the papers [19,20,11,6,7]. The main definitions concerning this paper see in [9].…”

“…We prove that l r = l r ′ , for r, r ′ ∈ I j (C). Consider the restriction τ ′ of τ on the perfect subcode By (7) and Lemma 2 we see that τ ′ fixes any element (r, 0) of I 0 (M (C, D µ )), so we have that l r is equal to 0 for all r ∈ I 0 (C).…”

Classification of the <inline-formula> <tex-math notation="LaTeX">$Z_{2}Z_{4}$ </tex-math></inline-formula>-Linear Hadamard Codes and Their Automorphism Groups

“…The permutation automorphism group of Z 2 Z 4linear (extended) 1-perfect codes has been studied in [22], [17]. The permutation automorphism group of (nonlinear) binary 1-perfect codes has also been studied before, obtaining some partial results [13], [12], [1], [9]. For Z 2 Z 4 -additive Hadamard codes with α = 0, the permutation automorphism group was characterized in [21], while the monomial automorphism group as well as the permutation automorphism group of the corresponding Z 2 Z 4 -linear codes had not been studied yet.…”