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code loops; symplectic cubic spaces; combinatorial polarization; binary linear codes; divisible codes
The existence and uniqueness (up to equivalence defined below) of code loops was first established by R. Griess in [3]. Nevertheless, the explicit construction of code loops remained open until T. Hsu introduced the notion of symplectic cubic spaces and their Frattini extensions, and pointed out how the construction of code loops followed from the (purely combinatorial) result of O. Chein and E. Goodaire contained in [2]. Within this paper, we focus on their combinatorial construction and prove a more general result 2.1 using the language of derived forms.
[1] Aschbacher M.: Sporadic Groups. Cambridge Tracts in Mathematics 104 (1994), Cambridge University Press. MR 1269103 | Zbl 0804.20011
[2] Chein O., Goodaire E.: Moufang loops with a unique nonidentity commutator (associator, square). J. Algebra 130 (1990), 369-384. MR 1051308 | Zbl 0695.20040
[3] Griess R.L., Jr.: Code loops. J. Algebra 100 (1986), 224-234. MR 0839580 | Zbl 0589.20051
[4] Hsu T.: Moufang loops of class $2$ and cubic forms. Math. Proc. Camb. Phil. Soc., to appear. MR 1735310 | Zbl 0962.20046
[5] Vojtěchovský P.: Derived Forms and Binary Linear Codes. Mathematics Report Number M99-10, Department of Mathematics, Iowa State University.
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