Ja n 20 07 COMBINATORIAL ASPECTS OF CODE LOOPS

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. Throughout this paper, let F = {0, 1} be the two-element field, and let V be a finite-dimensional vector space over F. For v ∈ V , let |v| denote the number of non-zero coordinates of v—the weight of v. When w is another vector in V , let v * w denote the vector whose ith coordinate is non-zero if and only if the ith coordinate of both v and w is non-zero. A binary linear code C ≤ V is said to be of level r if r is the biggest integer such that 2 r divides the weight of every codeword of C. We write lev(C) = r. A code C is doubly even if lev(C) ≥ 2. For the rest of this section, let C be a doubly even code. becomes a Moufang loop, a code loop of C. R. Griess shows in [3] that every C admits a factor set ϕ, and thus that there is a code loop for every doubly even code C. Moreover, when ϕ, ψ are two factor sets for C, then they are equivalent in the sense that the second derived form (ϕ + ψ) 2 is the zero mapping. (See section 2 for the definition of derived forms.) Note that a loop L is a code loop of C if there is a two-element central subgroup Z ≤ Z(L) such that L/Z is isomorphic to C as an elementary abelian 2-group. The following ideas are due to T. Hsu [4]. Let L be a code loop of C.