Abstract: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.
Keywords: code loops, symplectic cubic spaces, combinatorial polarization, binary linear codes, divisible codes
AMS Subject Classification: Primary 05A19; Secondary 94B05, 20N05