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Nordstrom-Robinson code and \(A_7\)-geometry. (English) Zbl 1134.94014
The Nordstrom-Robinson code by A. W. Nordstrom and J. P. Robinson [Inf. Control 11, 613–616 (1967; Zbl 0157.26003)] is a nonlinear \((16, 2^8, 6)\)-code over \(\mathbb{F}_2\). A simple construction for the Nordstrom-Robinson code is given by G. D. Forney jun., N. J. A. Sloane and M.D. Trott [Coding and quantization. DIMACS/IEEE workshop held at the Princeton University, NJ, USA, October 19-21, 1992. Providence, RI: American Mathematical Society. DIMACS, Ser. Discrete Math. Theor. Comput. Sci. 14, 19–26 (1993; Zbl 0804.94021)]. The Nordstrom-Robinson code admits a huge automorphism group of order \(| M_{24}| /(759\cdot 8)\). This automorphism group is a semidirect product of an elementary abelian group of order \(16\) and the alternating group \(A_7\). In this paper the sporadic \(A_7\)-geometry is constructed from all possible Fano planes on a set of seven elements. From this construction an elementary abelian group \(V\) of order \(16\) and the semidirect product \(G=VA\), where \(A\cong A_7\), is obtained. In the last part the author derives the Nordstrom-Robinson code from the construction of the geometry \(A_7\) and proves that the group \(G\) is the full automorphism group of the Nordstrom-Robinson code.

MSC:
94B60 Other types of codes
51E22 Linear codes and caps in Galois spaces
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