On the Stabilizers of the Minimum-Weight Codewords of the Binary Codes from Triangular Graphs

Abstract

The stabilizers of the minimum-weight codewords of the binary codes obtained from the strongly regular graphs \(T(n)\) defined by the primitive rank-\(3\) action of the alternating groups \(A_n\), where \(n \geq 5\), on \(\Omega^{(2)}\), the set of duads of \(\Omega = \{1,2,\ldots,n\}\) are examined. For a codeword \(w\) of minimum-weight in the binary code \(C\) obtained as stated above, from an adjacency matrix of the triangular graph \(T(n)\) defined by the primitive rank-3 action of the alternating groups \(A_n\) where \(n \geq 5\), on \(\Omega^{(2)}\), the set of duads of \(\Omega = \{1,2,\ldots,n\}\), we determine the stabilizer \(Aut(C)_w\) in \(Aut(C)\) and show that \(Aut(C)_w\) is a maximal subgroup of \(Aut(C)\).