Code loops
Recall the definitions: 1) Let \(V\) be a subspace of \(P(\Omega)\), the power set of a finite set \(\Omega\) regarded as a vector space over \(F_ 2\). We say that \(V\) is doubly even if \(A\in V\) implies \(| A| \equiv 0\pmod 4\). 2) Given a doubly even binary code of order \(2^ n\), we define a code loop as a certain Moufang loop of order \(2^{n+1}\). 3) Let \(V\leq P(\Omega)\) be doubly even. A factor set is a function \(\phi\colon V\times V\to F_ 2\) such that (S) \(\phi(x,x)=1/4| x|\); (C) \(\phi(x,y)+\phi(y,x)=1/2| x\cap y|\); (A) \(\phi(x,y)+\phi(y,z)+\phi(x+y,z)+\phi(x,y+z)=| x\cap y\cap z|\), for all \(x,y,z\in V\). The main result of this paper is the theorem: Given a doubly even subspace \(V\leq P(\Omega)\), a factor set exists. In fact, if \(n=\dim V\), there are \(2^{2^ n-n-1}\) of them. Any two are equivalent.
scientific article; zbMATH DE number 3946473